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Sophia leyson
https://www.cuemath.com/algebra/types-of-functions/
This will share or explain to you the different types and their meaning of functions

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The types of functions are defined on the basis of the domain, range, and function expression. The expression used to write the function is the prime defining factor for a function. Along with expression, the relationship between the elements of the domain set and the range set also accounts for the type of function. The classification of functions helps to easily understand and learn the different types of functions.

Every mathematical expression which has an input value and a resulting answer can be conveniently presented as a function. Here we shall learn about the types of functions and their definition, examples.

What are the Different Types of Functions?

The function y = f(x) is classified into different types of functions, based on factors such as the domain and range of a function, and the function expression. The functions have a domain x value that is referred as input. The domain value can be a number, angle, decimal, fraction. Similarly, the y value or the f(x) value (is generally a numeric value) is the range. The types of functions have been classified into the following four types.

Based on the Set Elements

Based on Equation

Based on Range

Based on Domain

Representation of Functions

There are three different forms of representation of functions. The functions need to be represented to showcase the domain values and the range values and the relationship between them. The functions can be represented with the help of Venn diagrams, graphical formats, and roster forms. The details of each of the forms of representation are as follows.

Venn Diagram: The Venn diagram is an important format for representing the function. The Venn diagrams are usually presented as two circles with arrows connecting the element in each of the circles. The domain is presented in one circle and the range values are presented in another circle. And the function defines the arrows, and how the arrows connect the different elements in the two circles.

Graphical Form: Functions are easy to understand if they are represented in the graphical form with the help of the coordinate axes. Representing the function in graphical form, helps us to understand the changing behavior of the functions if the function is increasing or decreasing. The domain of the function - the x value is represented along the x-axis, and the range or the f(x) value of the function is plotted with respect to the y-axis.

Roster Form: Roster notation of a set is a simple mathematical representation of the set in mathematical form. The domain and range of the function are represented in flower brackets with the first element of a pair representing the domain and the second element representing the range. Let us try to understand this with the help of a simple example. For a function of the form f(x) = x ², the function is represented as {(1, 1), (2, 4), (3, 9), (4, 16)}. Here the first element is the domain or the x value and the second element is the range or the f(x) value of the function.

List of Types of Functions

The types of functions are classified further to help for easy understanding and learning. The types of functions have been further classified into four different types, and are presented as follows.

Based on Elements

1. One One Function

2. Many One Function

3. Onto Function

4. One One and Onto Function

5. Into Function

6. Constant Function

Based on Equation

1. Identity Function

2. Linear Function

3. Quadratic Function

4. Cubic Function

5. Polynomial Functions

Based on the Range

1. Modulus Function

2. Rational Function

3. Signum Function

4. Even and Odd Functions

5. Periodic Functions

6. Greatest Integer Function

7. Inverse Function

8. Composite Functions

Based on the Domain

1. Algebraic Functions

2. Trigonometric Functions

3. Logarithmic Functions

Types of Functions - Based on Set Elements

These types of functions are classified based on the number of relationships between the elements in the domain and the codomain. The different types of functions based on set elements are as follows.

One One Function

A one-to-one function is defined by f: A → B such that every element of set A is connected to a distinct element in set B. The one-to-one function is also called an injective function. Here every element of the domain has a distinct image or co-domain element for the given function.

Many to One Function

A many to one function is defined by the function f: A → B, such that more than one element of the set A are connected to the same element in the set B. In a many to one function, more than one element has the same co-domain or image. If a many to one function, in the codomain, is a single value or the domain element are all connected to a single element, then it is called a constant function.

Onto Function

In an, onto function, every codomain element is related to the domain element. For a function defined by f: A → B, such that every element in set B has a pre-image in set A. The onto function is also called a subjective function.

One One and Onto Function (Bijection)

A function that is both a one and onto function is called a bijective function. Here every element of the domain is connected to a distinct element in the codomain and every element of the codomain has a pre-image. Also in other words every element of set A is connected to a distinct element in set B, and there is not a single element in set B which has been left out.

Into Function

Into function is exactly opposite in properties to an onto function. Here there are certain elements in the co-domain that do not have any pre-image. The elements in set B are excess and are not connected to any elements in set A.

Constant Function

A constant function is an important form of a many to one function. In a constant function, all the domain elements have a single image. The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain(x value), the same range value of K is obtained for a constant function.

Types of Function - Based on Equation

The algebraic expressions are also functions and are based on the degree of the polynomial. The functions based on equations are classified into the following equations based on the degree of the variable 'x'.

1. The polynomial function of degree zero is called a Constant Function.

2. The polynomial function of degree one is called a Linear Function.

3. The polynomial function of degree two is called a Quadratic Function.

4. The polynomial function of degree three is a Cubic Function.

Let us understand each of these functions in detail.

Identity Function

The identity function has the same domain and range. The identity function equation is f(x) = x, or y = x. The domain and range of the identity function is of the form {(1, 1), (2, 2), (3, 3), (4, 4).....(n, n)}.

The graph of the identity function is a straight line that is equally inclined to the coordinate axes and is passing through the origin. The identity function can take both positive and negative values and hence it is present in the first and the third quadrants of the coordinate axis.

Linear Function

A polynomial function having the first-degree equation is a linear function. The domain and range of a linear function is a real number, and it has a straight line graph. Equations such as y = x + 2, y = 3x, y = 2x - 1, are all examples of linear functions. The identity function of y = x can also be considered a linear function.

The general form of a linear function is f(x) = ax + b, is used to represent objective functions in linear programming problems. Here x, y are variables, and a, b are real numbers. Graphically the linear function can be represented by the equation of a line y = mx + c, where m is the slope of the line and c is the y-intercept of the line.

Quadratic Function

A quadratic function has a second-degree quadratic equation and it has a graph in the form of a curve. The general form of the quadratic function is f(x) = ax ² + bx + c, where a ≠ 0 and a, b, c are constant and x is a variable. The domain and range of the quadratic function is R.

The graph of a quadratic equation is a non-linear graph and is parabolic in shape. Examples of quadratic functions are f(x) = 3x ² + 5, f(x) = x ² - 3x + 2.

Cubic Function

A cubic function has an equation of degree three. The general form of a cubic function is f(x) = ax ³ + bx ² + cx +d, where a ≠ 0 and a, b, c, and d are real numbers & x is a variable. The domain and range of a cubic function is R.

The graph of a cubic function is more curved than the quadratic function. An example of cubic function is f(x) = 8x ³ + 5x ² + 3.

Polynomial Function

The general form of a polynomial function is f(x) = a nx n + a n-1x n-1 + a n-2x n-2+ ..... ax + b. Here n is a nonnegative integer and x is a variable. The domain and range of a polynomial function are R. Based on the power of the polynomial function, the functions can be classified as a quadratic function, cubic function, etc.

Types of Functions - Based on Range

Here the types of functions have been classified based on the range which is obtained from the given functions. The different types of functions based on the range are as follows.

Modulus Function

The modulus function gives the absolute value of the function, irrespective of the sign of the input domain value. The modulus function is represented as f(x) = |x|. The input value of 'x' can be a positive or a negative expression. The graph of a modulus function lies in the first and the second quadrants since the coordinates of the points on the graph are of the form (x, y), (-x, y).

Rational Function

A function that is composed of two functions and expressed in the form of a fraction is a rational function. A rational fraction is of the form f(x)/g(x), and g(x) ≠ 0. The functions used in this rational function can be an algebraic function or any other function. The graphical representation of these rational functions is similar to the asymptotes, since it does not touch the axis lines.

Signum Function

The signum function helps us to know the sign of the function and does not give the numeric value or any other values for the range. The range of the signum function is limited to {-1, 0, 1}. For the positive value of the domain, the signum function gives an answer of 1, for negative values the signum function gives an answer of -1, and for the 0 value of a domain, the image is 0. The signum function has wide application in software programming.

Even and Odd Function

The even and odd functions are based on the relationship between the input and the output values of the function. For the negative domain value, if the range is a negative value of the range of the original function, then the function is an odd function. And for the negative domain value, if the range is the same as that of the original function, then the function is an even function.

If f(-x) = f(x), for all values of x, then the function is an even function, and if f(-x) = -f(x), for all values of x, then the function is an odd function. An example of even functions are x ², Cosx, Secx, and an example of odd functions are x ³, Sinx, Tanx.

Periodic Function

The function is considered a periodic function if the same range appears for different domain values and in a sequential manner. The trigonometric functions can be considered periodic functions. For example, the function f(x) = Sinx, have a range [-1, 1] for the different domain values of x = nπ + (-1) n x. Similarly, we can write the domain and the range of the trigonometric functions and prove that the range shows up in a periodic manner.

Inverse function

The inverse of a function f(x) is denoted by f⁻¹(x). For inverse of a function the domain and range of the given function is changed as the range and domain of the inverse function. The inverse of a function can be prominently seen in algebraic functions and in inverse trigonometric functions. The domain of Sinx is R and its range is [-1, 1], and for Sin⁻¹ x the domain is [-1, 1] and the range is R. The inverse of a function exists, if it is a bijective function.

If a function f(x) = x ², then the inverse of the function is f⁻¹(x) = \(\sqrt x\).

Greatest Integer Function

The greatest integer function is also known as the step function. The greatest integer function rounds up the number to the nearest integer less than or equal to the given number. Clearly, the input variable x can take on any real value. However, the output will always be an integer. Also, all integers will occur in the output set. Thus, the domain of this function is real numbers R, while its range is integers (Z).

The greatest integer function graph is known as the step curve because of the step structure of the curve.The greatest integral function is denoted as f(x) = ⌊x⌋. For a function taking values from [1, 2), the value of f(x) is 1.

Composite Function

The composite functions are of the form of gof(x), fog(x), h(g(f(x))), and is made from the individual functions of f(x), g(x), h(x). The composite functions made of two functions have the range of one function forming the domain for another function. Let us consider a composite function fog(x), which is made up of two functions f(x) and g(x).

Here we write fog(x) = f(g(x)). The range of g(x) forms the domain for the function f(x). It can be considered as a sequence of two functions. If f(x) = 2x + 3 and g(x) = x + 1 we have fog(x) = f(g(x)) = f(x + 1) = 2(x + 1) + 3 = 2x + 5.

Types oF Functions - Based on Domain

Functions are used in all the other topics of maths. The functions have been classified based on the types of equations used to define the functions. The function equations generally have algebraic expressions, trigonometric functions, logarithms, exponents, and hence are named based on these domain values. The three broad types of functions based on domain value are as follows.

Algebraic Function

An algebraic function is helpful to define the various operations of algebra. The algebraic function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, division. An algebraic function is generally of the form of f(x) = a nx n + a n - 1x n - 1+ a n-2x n-2+ ....... ax + c.

The algebraic function can also be represented graphically. The algebraic function is also termed as a linear function, quadratic function, cubic function, polynomial function, based on the degree of the algebraic equation.

Trigonometric Functions

The trigonometric functions also have a domain and range similar to any other function. The six trigonometric functions are f(θ) = sinθ, f(θ) = cosθ, f(θ) = tanθ, f(θ) = secθ, f(θ) = cosecθ. Here the domain value θ is the angle and is in degrees or in radians. These trigonometric functions have been taken based on the ratio of the sides of a right-angle triangle, and are based on the Pythagoras theorem.

Further from these trigonometric functions, inverse trigonometric functions have also been derived. The domain of the inverse trigonometric function is a real number value and its range is an angle. The trigonometric functions and the inverse trigonometric functions are also sometimes referred to as periodic functions since the principal values are repeated.

Logarithmic Functions

Logarithmic functions have been derived from the exponential functions. The logarithmic functions are considered as the inverse of exponential functions. Logarithmic functions have a 'log' in the function and it has a base. The logarithmic function is of the form y = \(\log_ax \). Here the domain value is the input value of 'x' and is calculated using the Napier logarithmic table. The logarithmic function gives the number of exponential times to which the base has raised to obtain the value of x. The same logarithmic function can be expressed as an exponential function as x = a y.
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What are the Types of Functions in Maths?

The types of functions can be broadly classified into four types.

Based on Element: One to one Function, many to one function, onto function, one to one and onto function, into function.

Based on Domain: Algebraic Functions, Trigonometry functions, logarithmic functions.

Based on Range: Modulus function, rational function, signum function, even and odd function, greatest integer function.

Based on Equations: Identity function, linear function, quadratic function, cubic function, polynomial function.

How do you Determine the Types of Functions?

The types of functions can be determined based on the domain, range, and functional equation. Further classifying functions into types of functions helps to group and easily understand each of the types of functions. All the trigonometric functions can be grouped under periodic functions. Functions such as identity function, linear function, the quadratic, cubic function can be grouped under polynomial functions. Thus the function equation y = f(x) is helpful to define the type of function.

What Type of Functions is Always Continuous?

The functions which have a range for every input value of the domain can be termed as a continuous function. The simplest example of a continuous function is the identity function which has equal domain and range value. Other examples of continuous functions are the trigonometric sine function and cosine functions.

What are the Various Forms of Representation of Functions?

The functions are generally represented in the form of an equation y = f(x), where x is the domain and y or f(x) is the range of the function. The functions with the domain and range elements are also represented as venn diagrams or as roster form. In roster form the domain and range of the function are represented as {\((x_1, f(x_1)), (x_2, f(x_2)), (x_3, f(x_3))\)}.

What are the other math applications of Types of Functions?

The types of functions have enormous applications in algebra, trigonometry, logarithms, exponents. All the algebraic expressions can be counted as functions as it has an input domain value of x and the output range, which is the answer of the algebraic function. The trigonometric functions also have the angle value as the domain and range value.

Can any Math Equation be Considered a Function?

Yes. A math equation that is not equal to zero can be considered as a function. Alternatively, a math equation with two variables where one variable can be taken as a domain and the other variable can be taken as the range, can be called a function.

What are the applications of types of functions?

The types of functions have numerous applications in physics, engineering, computer sciences, artificial intelligence. All of these fields aim at connecting one set of data points (domain) to another set of data points(range). Also, the functions help in representing the huge set of data points in a simple mathematical expression of the formal y = f(x).

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Types of Functions - Types, Definition, Examples

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The types of functions are defined on the basis of the domain, range, and function expression. The expression used to write the function is the prime defining factor for a function. Along with expression, the relationship between the elements of the domain set and the range set also accounts for the type of function.

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Sophia Leyson https://ocw.bib.upct.es/pluginfile.php/8765/mod_resource/content/1/The_role_of_Mathematics_in_Business_studies.pdf
This is a PDF file that will help you comprehend the uses or importance of General Math in Business

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https://revisionmaths.com/advanced-level-maths-revision/pure-maths/algebra/functions

This website will give a look on the basic definition of functions and the different types of it.
Rojas, Chris Troy S.

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https://www.britannica.com/science/function-mathematics
OwenHarrison_This is the general definition of functions that will help give you an insight to what it is

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Sophia Leyson
https://study.com/academy/lesson/critical-thinking-and-logic-in-mathematics.html
This is a full webpage that will help you review on lessons on Mathematics Logic, you can type in any lesson and review yourself or with friends

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Critical Thinking and Logic in Mathematics - Video & Lesson Transcript | Study.com

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Mathematics involves logic and critical thinking to make connections and draw conclusions. Explore how to use logic, propositions, true or false,…

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https://en.wikipedia.org/wiki/Mathematical_logicv

Galing, Jephthah V.

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Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive powe…

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https://www.toppr.com/guides/maths/relations-and-functions/types-of-functions/

Nalaza, John Paul L.

This will guide you in defining what functions are in mathematics and show you examples of it.

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https://mathinsight.org/definition/function
Jared Jiongco_This gives the technical definition of a function and describes what is in a set of function

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A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

This means that if the object $x$ is in the set of inputs (called the domain) then a function $f$ will map the object $x$ to exactly one object $f(x)$ in the set of possible outputs (called the codomain).

The notion of a function is easily understood using the metaphor of a function machine that takes in an object for its input and, based on that input, spits out another object as its output.

A function is more formally defined given a set of inputs $X$ ( domain) and a set of possible outputs $Y$ ( codomain) as a set of ordered pairs $(x,y)$ where $x \in X$ (confused?) and $y \in Y$, subject to the restriction that there can be only one ordered pair with the same value of $x$. We can write the statement that $f$ is a function from $X$ to $Y$ using the function notation $f: X \to Y$.

For more information about functions, check out the function machine or the function examples.

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A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. This means that if the object $x$ is in the set of inputs (called the domain) then a function $f$ will map the object $x$ to exactly one object $f(x)$ in the set of possible outputs (called the codomain).

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https://www.vedantu.com/maths/business-mathematics
OwenHarrison_this will show you the topics covered in general mathematics

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https://testbook.com/learn/maths-types-of-functions/
Gabinete, Joseph Christian R. Gabinete

this webpage will explain the definition of each type of functions and helps you understand what they are.

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Types of Functions: Definition of Types of Functions with Example

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Types of Functions: Check the Types of Functions in Mathematics with Examples One-One, Many-One, bijective, etc. Also get their definition & representation

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https://www.khanacademy.org/test-prep/sat/x0a8c2e5f:untitled-652/x0a8c2e5f:passport-to-advanced-math-lessons-by-skill/a/gtp--sat-math--article--function-notation--lesson#:~:text=A%20function%20takes%20an%20input,right%20parenthesis%20is%20the%20output.

SteffiAnnRosales_ Functions take input and produce output. The input can be numeric, expression, or another function.

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Function Notation | Lesson (article) | Khan Academy

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https://byjus.com/jee/functions-and-its-types/

Loket, Brian Angelo C.

This website show us the different contents of Function

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Functions - Definition, Types, Domain Range and Video Lesson

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Functions and different types of functions are explained here along with solved examples. Visit BYJU’S to learn about the various functions in mathematics in detail with a video lesson and download functions and types of functions PDF for free.

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https://www.britannica.com/science/function-mathematics
DarleneSerranilla_here you will find the definition of functions and more info about what functions is all about.

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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Sophia Leyson
https://www.britannica.com/science/function-mathematics
what is Function, This will give you a quick explanation on how to understand Functions

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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https://mathigon.org/world/Logic_and_Paradoxes
OwenHarrison_this link will help guide you through the basics of logic such as its definition and a few examples of what will be done in the topic

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Solving Functions: Tables, Graphs, Equations (Simplifying Math)

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Solving Functions: Tables, Graphs, Equations (Simplifying Math)

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https://www.youtube.com/watch?v=ldt1CzensyA

Galing, Jephthah V.

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Solving Functions: Tables, Graphs, Equations (Simplifying Math)

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https://www.zambianguardian.com/importance-of-business-math-in-acco

Galing, Jephthah V.

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The importance of business math in accounting has been significantly noted around the world. Human resources and public administration do not necessarily need mathematics compared to computer science. Even though in the course curriculum it's possible to find some accounting principles and studies. Business mathematics in accounting for Commercial, Government, and Business organizations use mathematics in accounting for checks and balances, financial evaluations, financial accounting, and calculations.

Importance of business mathematics in accounting

Here's the role of mathematics in the accounting field and the importance of business math in accounting

1. Checks and balances

Business math in accounting is used when accountants want to perform a round count to know what's missing. Sometimes this happens when analyzing inventory as well as cash sales. Checks and balances in and public administration have been an important principle of accounting that requires mathematics in separate government branches to support proper actions by accounting through diverse checks and balances applied in financial departments. Hence financial checks and balances can also be said to be the importance of mathematics in business Administration and bureaucracy.

2. Financial Evaluations

Accounting evaluations are another study to find and note the importance of business math in accounting. The evaluation process especially in business actually means the measurement of how things get done in the organization through mathematical means and procedures.

Evaluation is seen as a process for the measurement of most organizational operations. The importance of business math in performing accounting helps also for advocating the idea of developing an accounts theory well-tailored to modern accounting essential standards.

3. Financial accounting

Financial accounting is a process of accounting that involves business math for different methods of documenting cognitive responses from the accounts department. It then summarizes them before taking the step of reporting the transactions.

The importance of business mathematics in accounting arising from businesses' everyday operations needs financial accounting and reflects the process of accounting on an accrual basis to promote accounting accrual "cash basis".

4. Calculations

Business math in accounting helps with necessary calculations that may arise from the need to properly ensures everything balances. To accounting department and business math, means to determine by implementing mathematically methods through the use of numbers or by combining or manipulating amounts. Calculations are mostly done with help of a gadget called a calculator. This helps with assessing any situation and figuring out the possible expected outcome.

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Importance of business math in accounting: Business mathematics

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The importance of business math in accounting has been significantly noted around the world. Human resources and public administration do not necessarily need mathematics compared to computer science. Even though in the course curriculum it's possible to find some accounting principles and studies.

# data[].attributes.body
https://byjus.com/maths/what-is-a-function/
The website includes some definitions and examples of different types of functions such as injective, surjective, and polynomial functions.

Pantas, Aicelle P.

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What is a Function? Definition, Types and Notation

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Learn what is a function along with its types and examples. A function in Maths is defined as a relation between two or more variables, whose graph can be plotted in cartesian plane.

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https://www.britannica.com/science/function-mathematics

Flores, Elisha D.
This link shows us the functions, types, examples, and facts of our topic "functions" in general mathematics.

# data[].attributes.attachment_link.readable_text
function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

# data[].attributes.attachment_link.title
Function | Definition, Types, Examples, & Facts

# data[].attributes.attachment_link.description
Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

# data[].attributes.body
https://mathinsight.org/definition/function
Dragon, Jandex Nicole P.
A simple explanation of what Function is to give better insight of what it is all about and to give idea how to solve its problems

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A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

This means that if the object $x$ is in the set of inputs (called the domain) then a function $f$ will map the object $x$ to exactly one object $f(x)$ in the set of possible outputs (called the codomain).

The notion of a function is easily understood using the metaphor of a function machine that takes in an object for its input and, based on that input, spits out another object as its output.

A function is more formally defined given a set of inputs $X$ ( domain) and a set of possible outputs $Y$ ( codomain) as a set of ordered pairs $(x,y)$ where $x \in X$ (confused?) and $y \in Y$, subject to the restriction that there can be only one ordered pair with the same value of $x$. We can write the statement that $f$ is a function from $X$ to $Y$ using the function notation $f: X \to Y$.

For more information about functions, check out the function machine or the function examples.

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A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. This means that if the object $x$ is in the set of inputs (called the domain) then a function $f$ will map the object $x$ to exactly one object $f(x)$ in the set of possible outputs (called the codomain).

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This website will elaborate on what functions are and how they work and it will briefly explain every question you have for Functions.
Hayden Villanueva.

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You appear to be on a device with a "narrow" screen width ( i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

A relation is a set of ordered pairs.

A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.

Example 1 The following relation is a function. \[\left\{ {\left( { - 1,0} \right)\,\,\,\,\left( {0, - 3} \right)\,\,\,\,\left( {2, - 3} \right)\,\,\,\,\left( {3,0} \right)\,\,\,\,\left( {4,5} \right)} \right\}\]

Show Solution

From these ordered pairs we have the following sets of first components ( i.e. the first number from each ordered pair) and second components ( i.e. the second number from each ordered pair).

\[{1^{{\mbox{st}}}}{\mbox{ components : }}\left\{ { - 1,0,2,3,4} \right\}\hspace{0.25in}\hspace{0.25in}{2^{{\mbox{nd}}}}{\mbox{ components : }}\left\{ {0, - 3,0,5} \right\}\]

For the set of second components notice that the "-3" occurred in two ordered pairs but we only listed it once.

To see why this relation is a function simply pick any value from the set of first components. Now, go back up to the relation and find every ordered pair in which this number is the first component and list all the second components from those ordered pairs. The list of second components will consist of exactly one value.

For example, let's choose 2 from the set of first components. From the relation we see that there is exactly one ordered pair with 2 as a first component,\(\left( {2, - 3} \right)\). Therefore, the list of second components ( i.e. the list of values from the set of second components) associated with 2 is exactly one number, -3.

Note that we don't care that -3 is the second component of a second ordered par in the relation. That is perfectly acceptable. We just don't want there to be any more than one ordered pair with 2 as a first component.

We looked at a single value from the set of first components for our quick example here but the result will be the same for all the other choices. Regardless of the choice of first components there will be exactly one second component associated with it.

Therefore, this relation is a function.

Example 2 The following relation is not a function. \[\left\{ {\left( {6,10} \right)\,\,\,\,\left( { - 7,3} \right)\,\,\,\,\left( {0,4} \right)\,\,\,\,\left( {6, - 4} \right)} \right\}\]

Show Solution

Don't worry about where this relation came from. It is just one that we made up for this example.

Here is the list of first and second components

\[{1^{{\mbox{st}}}}{\mbox{ components : }}\left\{ {6, - 7,0} \right\}\hspace{0.25in}\hspace{0.25in}{2^{{\mbox{nd}}}}{\mbox{ components : }}\left\{ {10,3,4, - 4} \right\}\]

From the set of first components let's choose 6. Now, if we go up to the relation we see that there are two ordered pairs with 6 as a first component : \(\left( {6,10} \right)\) and \(\left( {6, - 4} \right)\). The list of second components associated with 6 is then : 10, -4.

The list of second components associated with 6 has two values and so this relation is not a function.

Note that the fact that if we'd chosen -7 or 0 from the set of first components there is only one number in the list of second components associated with each. This doesn't matter. The fact that we found even a single value in the set of first components with more than one second component associated with it is enough to say that this relation is not a function.

As a final comment about this example let's note that if we removed the first and/or the fourth ordered pair from the relation we would have a function!

A function is an equation for which any \(x\) that can be plugged into the equation will yield exactly one \(y\) out of the equation.

Example 3 Determine which of the following equations are functions and which are not functions. Show All Solutions Hide All Solutions

Show Discussion

The "working" definition of function is saying is that if we take all possible values of \(x\) and plug them into the equation and solve for \(y\) we will get exactly one value for each value of \(x\). At this stage of the game it can be pretty difficult to actually show that an equation is a function so we'll mostly talk our way through it. On the other hand, it's often quite easy to show that an equation isn't a function.

So, we need to show that no matter what \(x\) we plug into the equation and solve for \(y\) we will only get a single value of \(y\). Note as well that the value of \(y\) will probably be different for each value of \(x\), although it doesn't have to be.

Let's start this off by plugging in some values of \(x\) and see what happens.

\[\begin{align*}x & = - 4:\hspace{0.25in} & y & = 5\left( { - 4} \right) + 1 = - 20 + 1 = - 19\\ x & = 0:\hspace{0.25in} & y & = 5\left( 0 \right) + 1 = 0 + 1 = 1\\ x & = 10:\hspace{0.25in} & y & = 5\left( {10} \right) + 1 = 50 + 1 = 51\end{align*}\]

So, for each of these values of \(x\) we got a single value of \(y\) out of the equation. Now, this isn't sufficient to claim that this is a function. In order to officially prove that this is a function we need to show that this will work no matter which value of \(x\) we plug into the equation.

Of course, we can't plug all possible value of \(x\) into the equation. That just isn't physically possible. However, let's go back and look at the ones that we did plug in. For each \(x\), upon plugging in, we first multiplied the \(x\) by 5 and then added 1 onto it. Now, if we multiply a number by 5 we will get a single value from the multiplication. Likewise, we will only get a single value if we add 1 onto a number. Therefore, it seems plausible that based on the operations involved with plugging \(x\) into the equation that we will only get a single value of \(y\) out of the equation.

So, this equation is a function.

Again, let's plug in a couple of values of \(x\) and solve for \(y\) to see what happens.

\[\begin{align*}x & = - 1:\hspace{0.25in} & y = {\left( { - 1} \right)^2} + 1 = 1 + 1 = 2\\ x & = 3:\hspace{0.25in} & y = {\left( 3 \right)^2} + 1 = 9 + 1 = 10\end{align*}\]

Now, let's think a little bit about what we were doing with the evaluations. First, we squared the value of \(x\) that we plugged in. When we square a number there will only be one possible value. We then add 1 onto this, but again, this will yield a single value.

So, it seems like this equation is also a function.

Note that it is okay to get the same \(y\) value for different \(x\)'s. For example,

\[x = - 3:\hspace{0.25in}y = {\left( { - 3} \right)^2} + 1 = 9 + 1 = 10\]

We just can't get more than one \(y\) out of the equation after we plug in the \(x\).

As we've done with the previous two equations let's plug in a couple of value of \(x\), solve for \(y\) and see what we get.

\[\begin{align*}x & = 3:\hspace{0.25in} & {y^2} & = 3 + 1 = 4\hspace{0.25in}\Rightarrow \hspace{0.25in} y = \pm 2\\ x & = - 1:\hspace{0.25in} & {y^2} & = - 1 + 1 = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}y = 0\\ x & = 10:\hspace{0.25in} & {y^2} & = 10 + 1 = 11\hspace{0.25in} \Rightarrow \hspace{0.25in}y = \pm \sqrt {11} \end{align*}\]

Now, remember that we're solving for \(y\) and so that means that in the first and last case above we will actually get two different \(y\) values out of the \(x\) and so this equation is NOT a function.

Note that we can have values of \(x\) that will yield a single \(y\) as we've seen above, but that doesn't matter. If even one value of \(x\) yields more than one value of \(y\) upon solving the equation will not be a function.

What this really means is that we didn't need to go any farther than the first evaluation, since that gave multiple values of \(y\).

With this case we'll use the lesson learned in the previous part and see if we can find a value of \(x\) that will give more than one value of \(y\) upon solving. Because we've got a in the problem this shouldn't be too hard to do since solving will eventually mean using the square root property which will give more than one value of \(y\).

\[x = 0:\hspace{0.25in}{0^2} + {y^2} = 4\hspace{0.25in} \Rightarrow \hspace{0.25in}{y^2} = 4\hspace{0.25in} \Rightarrow \hspace{0.25in} y = \pm \,2\]

So, this equation is not a function. Recall, that from the previous section this is the equation of a circle. Circles are never functions.

\[y = {x^2} - 5x + 3\]

Example 4 Given \(f\left( x \right) = {x^2} - 2x + 8\) and \(g\left( x \right) = \sqrt {x + 6} \) evaluate each of the following.

1. \(f\left( 3 \right)\) and \(g\left( 3 \right)\)

2. \(f\left( { - 10} \right)\) and \(g\left( { - 10} \right)\)

3. \(f\left( 0 \right)\)

4. \(f\left( t \right)\)

5. \(f\left( {t + 1} \right)\) and \(f\left( {x + 1} \right)\)

6. \(f\left( {{x^3}} \right)\)

7. \(g\left( {{x^2} - 5} \right)\)

Show All Solutions Hide All Solutions
a \(f\left( 3 \right)\) and \(g\left( 3 \right)\) Show Solution

Okay we've got two function evaluations to do here and we've also got two functions so we're going to need to decide which function to use for the evaluations. The key here is to notice the letter that is in front of the parenthesis. For \(f\left( 3 \right)\) we will use the function \(f\left( x \right)\) and for \(g\left( 3 \right)\) we will use \(g\left( x \right)\). In other words, we just need to make sure that the variables match up.

Here are the evaluations for this part.

\[\begin{align*}f\left( 3 \right) & = {\left( 3 \right)^2} - 2\left( 3 \right) + 8 = 9 - 6 + 8 = 11\\ g\left( 3 \right) & = \sqrt {3 + 6} = \sqrt 9 = 3\end{align*}\]

b \(f\left( { - 10} \right)\) and \(g\left( { - 10} \right)\) Show Solution

This one is pretty much the same as the previous part with one exception that we'll touch on when we reach that point. Here are the evaluations.

\[f\left( { - 10} \right) = {\left( { - 10} \right)^2} - 2\left( { - 10} \right) + 8 = 100 + 20 + 8 = 128\]

Make sure that you deal with the negative signs properly here. Now the second one.

\[g\left( { - 10} \right) = \sqrt { - 10 + 6} = \sqrt { - 4} \]

We've now reached the difference. Recall that when we first started talking about the definition of functions we stated that we were only going to deal with real numbers. In other words, we only plug in real numbers and we only want real numbers back out as answers. So, since we would get a complex number out of this we can't plug -10 into this function.

Not much to this one.

\[f\left( 0 \right) = {\left( 0 \right)^2} - 2\left( 0 \right) + 8 = 8\]

Again, don't forget that this isn't multiplication! For some reason students like to think of this one as multiplication and get an answer of zero. Be careful.

The rest of these evaluations are now going to be a little different. As this one shows we don't need to just have numbers in the parenthesis. However, evaluation works in exactly the same way. We plug into the \(x\)'s on the right side of the equal sign whatever is in the parenthesis. In this case that means that we plug in \(t\) for all the \(x\)'s.

Here is this evaluation.

\[f\left( t \right) = {t^2} - 2t + 8\]

Note that in this case this is pretty much the same thing as our original function, except this time we're using \(t\) as a variable.

e \(f\left( {t + 1} \right)\) and \(f\left( {x + 1} \right)\) Show Solution

Now, let's get a little more complicated, or at least they appear to be more complicated. Things aren't as bad as they may appear however. We'll evaluate \(f\left( {t + 1} \right)\) first. This one works exactly the same as the previous part did. All the \(x\)'s on the left will get replaced with \(t + 1\). We will have some simplification to do as well after the substitution.

\[\begin{align*}f\left( {t + 1} \right) & = {\left( {t + 1} \right)^2} - 2\left( {t + 1} \right) + 8\\ & = {t^2} + 2t + 1 - 2t - 2 + 8\\ & = {t^2} + 7\end{align*}\]

Be careful with parenthesis in these kinds of evaluations. It is easy to mess up with them.

Now, let's take a look at \(f\left( {x + 1} \right)\). With the exception of the \(x\) this is identical to \(f\left( {t + 1} \right)\) and so it works exactly the same way.

\[\begin{align*}f\left( {x + 1} \right) & = {\left( {x + 1} \right)^2} - 2\left( {x + 1} \right) + 8\\ & = {x^2} + 2x + 1 - 2x - 2 + 8\\ & = {x^2} + 7\end{align*}\]

Do not get excited about the fact that we reused \(x\)'s in the evaluation here. In many places where we will be doing this in later sections there will be \(x\)'s here and so you will need to get used to seeing that.

Again, don't get excited about the \(x\)'s in the parenthesis here. Just evaluate it as if it were a number.

\[\begin{align*}f\left( {{x^3}} \right) & = {\left( {{x^3}} \right)^2} - 2\left( {{x^3}} \right) + 8\\ & = {x^6} - 2{x^3} + 8\end{align*}\]

One more evaluation and this time we'll use the other function.

\[\begin{align*}g\left( {{x^2} - 5} \right) & = \sqrt {{x^2} - 5 + 6} \\ & = \sqrt {{x^2} + 1} \end{align*}\]

Example 5 Given, \[g\left( t \right) = \left\{ {\begin{array}{*{20}{l}}{3{t^2} + 4}&{{\mbox{if }}t \le - 4}\\{10}&{{\mbox{if }} - 4 < t \le 15}\\{1 - 6t}&{{\mbox{if }}t > 15}\end{array}} \right.\]

evaluate each of the following.

1. \(g\left( { - 6} \right)\)

2. \(g\left( { - 4} \right)\)

3. \(g\left( 1 \right)\)

4. \(g\left( {15} \right)\)

5. \(g\left( {21} \right)\)

Show All Solutions Hide All Solutions
Show Discussion

Before starting the evaluations here let's notice that we're using different letters for the function and variable than the ones that we've used to this point. That won't change how the evaluation works. Do not get so locked into seeing \(f\) for the function and \(x\) for the variable that you can't do any problem that doesn't have those letters.

Now, to do each of these evaluations the first thing that we need to do is determine which inequality the number satisfies, and it will only satisfy a single inequality. When we determine which inequality the number satisfies we use the equation associated with that inequality.

So, let's do some evaluations.

In this case -6 satisfies the top inequality and so we'll use the top equation for this evaluation.

\[g\left( { - 6} \right) = 3{\left( { - 6} \right)^2} + 4 = 112\]

Now we'll need to be a little careful with this one since -4 shows up in two of the inequalities. However, it only satisfies the top inequality and so we will once again use the top function for the evaluation.

\[g\left( { - 4} \right) = 3{\left( { - 4} \right)^2} + 4 = 52\]

In this case the number, 1, satisfies the middle inequality and so we'll use the middle equation for the evaluation. This evaluation often causes problems for students despite the fact that it's actually one of the easiest evaluations we'll ever do. We know that we evaluate functions/equations by plugging in the number for the variable. In this case there are no variables. That isn't a problem. Since there aren't any variables it just means that we don't actually plug in anything and we get the following,

\[g\left( 1 \right) = 10\]

Again, like with the second part we need to be a little careful with this one. In this case the number satisfies the middle inequality since that is the one with the equal sign in it. Then like the previous part we just get,

\[g\left( {15} \right) = 10\]

Don't get excited about the fact that the previous two evaluations were the same value. This will happen on occasion.

For the final evaluation in this example the number satisfies the bottom inequality and so we'll use the bottom equation for the evaluation.

\[g\left( {21} \right) = 1 - 6\left( {21} \right) = - 125\]

The domain of an equation is the set of all \(x\)'s that we can plug into the equation and get back a real number for \(y\). The range of an equation is the set of all \(y\)'s that we can ever get out of the equation.

Example 6 Determine the domain of each of the following functions.

1. \(\displaystyle g\left( x \right) = \frac{{x + 3}}{{{x^2} + 3x - 10}}\)

2. \(f\left( x \right) = \sqrt {5 - 3x} \)

3. \(\displaystyle h\left( x \right) = \frac{{\sqrt {7x + 8} }}{{{x^2} + 4}}\)

4. \(\displaystyle R\left( x \right) = \frac{{\sqrt {10x - 5} }}{{{x^2} - 16}}\)

Show All Solutions Hide All Solutions
Show Discussion

The domains for these functions are all the values of \(x\) for which we don't have division by zero or the square root of a negative number. If we remember these two ideas finding the domains will be pretty easy.

a \(\displaystyle g\left( x \right) = \frac{{x + 3}}{{{x^2} + 3x - 10}}\) Show Solution

So, in this case there are no square roots so we don't need to worry about the square root of a negative number. There is however a possibility that we'll have a division by zero error. To determine if we will we'll need to set the denominator equal to zero and solve.

\[{x^2} + 3x - 10 = \left( {x + 5} \right)\left( {x - 2} \right) = 0\hspace{0.25in}x = - 5,\,\,x = 2\]

So, we will get division by zero if we plug in \(x = - 5\) or \(x = 2\). That means that we'll need to avoid those two numbers. However, all the other values of \(x\) will work since they don't give division by zero. The domain is then,

\[{\mbox{Domain : All real numbers except }}x = - 5{\mbox{ and }}x = 2\]

b \(f\left( x \right) = \sqrt {5 - 3x} \) Show Solution

In this case we won't have division by zero problems since we don't have any fractions. We do have a square root in the problem and so we'll need to worry about taking the square root of a negative numbers.

This one is going to work a little differently from the previous part. In that part we determined the value(s) of \(x\) to avoid. In this case it will be just as easy to directly get the domain. To avoid square roots of negative numbers all that we need to do is require that

\[5 - 3x \ge 0\]

This is a fairly simple linear inequality that we should be able to solve at this point.

\[5 \ge 3x\hspace{0.25in} \Rightarrow \hspace{0.25in}x \le \frac{5}{3}\]

The domain of this function is then,

\[{\mbox{Domain : }}x \le \frac{5}{3}\]

c \(\displaystyle h\left( x \right) = \frac{{\sqrt {7x + 8} }}{{{x^2} + 4}}\) Show Solution

In this case we've got a fraction, but notice that the denominator will never be zero for any real number since is guaranteed to be positive or zero and adding 4 onto this will mean that the denominator is always at least 4. In other words, the denominator won't ever be zero. So, all we need to do then is worry about the square root in the numerator.

To do this we'll require,

\[\begin{align*}7x + 8 & \ge 0\\ 7x & \ge - 8\\ x & \ge - \frac{8}{7}\end{align*}\]

Now, we can actually plug in any value of \(x\) into the denominator, however, since we've got the square root in the numerator we'll have to make sure that all \(x\)'s satisfy the inequality above to avoid problems. Therefore, the domain of this function is

\[{\mbox{Domain :}}\,\,x \ge - \frac{8}{7}\]

d \(\displaystyle R\left( x \right) = \frac{{\sqrt {10x - 5} }}{{{x^2} - 16}}\) Show Solution

In this final part we've got both a square root and division by zero to worry about. Let's take care of the square root first since this will probably put the largest restriction on the values of \(x\). So, to keep the square root happy ( i.e. no square root of negative numbers) we'll need to require that,

\[\begin{align*}10x - 5 & \ge 0\\ 10x & \ge 5\\ x & \ge \frac{1}{2}\end{align*}\]

So, at the least we'll need to require that \(x \ge \frac{1}{2}\) in order to avoid problems with the square root.

Now, let's see if we have any division by zero problems. Again, to do this simply set the denominator equal to zero and solve.

\[{x^2} - 16 = \left( {x - 4} \right)\left( {x + 4} \right) = 0\hspace{0.25in} \Rightarrow \hspace{0.25in} x = - 4,\,\,x = 4\]

Now, notice that \(x = - 4\) doesn't satisfy the inequality we need for the square root and so that value of \(x\) has already been excluded by the square root. On the other hand, \(x = 4\) does satisfy the inequality. This means that it is okay to plug \(x = 4\) into the square root, however, since it would give division by zero we will need to avoid it.

The domain for this function is then,

\[{\mbox{Domain : }}x \ge \frac{1}{2}{\mbox{ except }}x = 4\]

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In this section we will formally define relations and functions. We also give a "working definition" of a function to help understand just what a function is. We introduce function notation and work several examples illustrating how it works. We also define the domain and range of a function.

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Rhyme Godfrey Anque_it shows here the meaning of business mathematics and its concept and explanation

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PhioGodinez_This website shows the different types of functions and its definitions

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Functions - Definition, Types, Domain Range and Video Lesson

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Functions and different types of functions are explained here along with solved examples. Visit BYJU’S to learn about the various functions in mathematics in detail with a video lesson and download functions and types of functions PDF for free.

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dela Rosa, Rob Aileron R,
In this website you will be provided with different examples of functions as well as their explanations to help you understand functions better.

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Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x. Mapping or transformation is used to denote a function in math. These functions are usually denoted by letters such as f, g, and h. The domain is defined as the set of all the values that the function can input while it can be defined. The range is all the values that come out as the output of the function involved. Co-domain is the set of values that have the potential of coming out as outputs of a function. Let us explore the world of functions in math.

What Are Functions?

A function is a process or a relation that associates each element 'a' of a non-empty set A , at least to a single element 'b' of another non-empty set B. A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math. f = {(a,b)| for all a ∈ A, b ∈ B}

A relation is said to be a function if every element of set A has one and only one image in set B.

A function is a relation from a non-empty set B such that the domain of a function is A and no two distinct ordered pairs in f have the same first element.

A function from A → B and (a,b) ∈ f, then f(a) = b, where 'b' is the image of 'a' under 'f' and 'a' is the preimage of 'b' under 'f'.

If there exists a function f: A → B, the set A is called the domain of the function f, and the set B is called its co-domain.

Examples of Functions

Whenever we say that a variable quantity y is a function of a variable quantity x, we mean to say that: y depends on x; the value of y is determined by the value of x. We can write this dependence as follows: y = f (x)

1. The area of a circle can be expressed in terms of its radius A = π r ². Area A is dependent on the radius r. In the language of functions in math, we say that A is a function of r.

2. The volume V of a sphere is a function of its radius. The dependence of V on r is given by V = 4/3 π r ³.

3. The acceleration a of a body of fixed mass m is a function of the force F applied on the body: a = F/m.

4. The power P dissipated in a resistor of fixed resistance R is a function of the current I passing through the resistor: \(P = {I^2}R\).

5. Suppose that a taxi charges a down payment of $10, and subsequently, $ 2 for every mile traveled. The taxi fare F is a function of the distance d traveled, and the dependence between the two quantities is given by F = 10 + 2d ( F is in dollars and d is in miles).

Representation of Functions in Math

The rule which specifies a function can come in many different forms based on how it is defined. They can be defined as piecewise-defined-functions or as formulas. When we define f(x) = √x, for x ≥ 0, then the inputs are the numbers that we provide and the 'taking square root' function accepts all non-negative real numbers and gets the output as f(x). Most often it is a formula, as in

\(\begin{equation*}
g(x) = \begin{cases}
2x & x < 0\\
x^2 & x \geq 0\\
\end{cases}
\end{equation*}\)

where the domain of g = all real numbers

A function in math can be represented as:

a set of ordered pairs

an arrow diagram

a table form

a graphical form

"f(x) = x ² " is the general manner to represent a function. It is said as f of x is equal to x square. This is represented as f = {(1,1), (2,4), (3,9)}. The domain and range of a function is given as D= {1, 2, 3}, R={1,4, 9}. Here is a representation of a function in math as an ordered pair.
Identification of a Function in Math

A function in math means a correspondence from one value x of the first set to another value y of the second set. This correspondence can be of the following four types. But every correspondence is not a function.

In the example below, only 1 - 1 and many to one are examples of a function because no two ordered pairs have the same first component and all elements of the first set are linked in them. So we say that in a function one input can result in only one output. If we are given any x then there is one and only one y that can be paired with that x. A function cannot be associated with two outputs. A curve drawn in a graph is represented as a function, then every vertical line intersects the curve in at most one point. This vertical line test helps us in determining whether the curve is a function or not.

Types of Functions

Functions in math have paramount importance and let us study different types of functions. We have four functions based on the mapping of elements from set A to set B.

f: A → B is said to be one-to-one or injective, if the images of distinct elements of A under f are distinct, i.e, for every a, b in A, f(a) = f(b), ⇒ a = b. Otherwise, it is many-to-one.

f: A → B is said to be onto, if every element of B is the image of some element of A under f, i.e, for every b ϵ B, there exists an element a in A such that f(a) = b. A function is onto if and only if the range of the function = B.

f: A → B is said to be one-to-one and onto or bijective, if f is both one-one and onto.

Composition of Functions

Let f : A → B and g: B→ C be two functions. Then the composition of f and g is denoted as f(g) and it is defined as the function f \(\circ\) g = f(g(x)) for x ∈ A. Let us consider an example of two functions f(x) and g(x). Let f(x) = (x+1) and g(x) = x ². Then f \(\circ\) g = f(x ²)

g \(\circ\) f = g(f(x))

g(f(x)) = g((x+1))

Thus we can conclude that the composition of functions is not commutative. f \(\circ\) g ≠ g \(\circ\) f

Algebra of Functions

The algebra of functions deals with the operations of functions. For the functions f(x) and g(x), where f: X → R and g: X → R, where x ϵ X, we have:

(f+g) (x) = f(x) + g(x)

(f-g) (x) = f(x) - g(x)

(f.g) (x) = f(x) .g(x)

(k f(x)) = k (f(x)), where k is a real number

(f/g)(x) = f(x) /g(x), where g(x) ≠ 0

Graphing Functions

The various types of functions are identity function, constant function, quadratic function, cubic function, cube root function, rational function, modulus function, and so on. Graphing functions is done by placing the independent variable on the x-axis and the dependent variable on the y-axis and plotting the points with coordinates (x,y) in the cartesian plane. All the points on the graph satisfy the equation y = f(x).

The identity function is the type of function which gives the same input as the output. It is expressed as, f(x) = x, where x ϵ R. For example, f(3) = 3 is an identity function.

A constant function is the type of function which gives the same value of output for any given input. It is expressed as, f(x) = c, where c is a constant. For example, f(x) = 2 is a constant function.

A Polynomial function is a type of function which can be expressed as a polynomial. It is expressed as,\(f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} +....... + a_{0}x^{0} \). For example, f(x) = 2x+5 is a polynomial function. f(x)= x³ - 6x²+ 11x - 6

A Quadratic function is a type of function which has the highest power 2 in the polynomial function. It is expressed as,f(x)=a(x-h)²+k. For example, f(x) = x2 + 1 is a quadratic function. The following figure shows the plot of a quadratic function.

A cubic function is a type of function which has the highest power 3 in the polynomial function.It is expressed as, f(x) = ax³ + bx² + cx + d, where a, b, c, d are constants. For example, f(x) = x³ + 4 is a cubic function. The following figure shows the plot of a cubic function.

A Rational function is a type of function which is derived from the ratio of two polynomial functions. It is expressed as, f(x) = P(x)/Q(x), such that P and Q are polynomial functions of x and Q(x) ≠ 0. For example, \(f(x) = \frac{x^2 + 2x + 1}{x^2 - 4}\) is a rational function.

Modulus function is a type of function which gives the absolute value of a number by giving its magnitude. It is expressed as, f(x) = |x| It can further be defined as:

\(\begin{equation*}
f(x) = \begin{cases}
x & x \geq 0\\
-x & x < 0\\
\end{cases}
\end{equation*}\)
Important Notes

A function in math means a correspondence from one value x of the first set A to another value y of the second set B. It relates inputs to outputs.

Functions in maths are the subsets of relations.

If A and B are finite sets such that n(A) = p and n(B) =q, then the total number of functions that exist from A → B is qp

Special Rules: a) A function must work for all the possible input values b) With each input value, it must have one relationship only. c) every element in the domain of 'f' has only an image.

Let us have a look at solved examples to know what are functions in math.
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FAQs on Functions in Math

What is a Function in Math?

A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math.

What are Parts of a Function?

The three basic concepts that help define any function are domain, range, and co-domain. The domain is defined as the set of all the values that the function can input while it can be defined. The range is all the values that come out as the output of the function involved. Co-domain is the set of values that have the potential of coming out as outputs of a function.

How Do You Write a Function in Math?

A function is a process or a relation that associates each element 'a' of a set A , to a single element 'b' of another set B. The function is represented as f: A → B. "f(x) = x ²" is the general manner to represent a function. For every x as the input value, the function of squaring is done and the output gets generated.

What is The Basic Concept of Function in Math?

A relation is said to be a function if every element of set A has one and only one image in set B. The function must work for all the possible input values and with each input value, it must have one relationship only. A function is a relation from a non-empty set B such that the domain of a function is A and no two distinct ordered pairs in f have the same first element.

How to do Transformations of Functions?

The different transformations are reflection, translation, rotation, and dilation. Follow the rules of the different transformations to find the transformations of functions. For example, f(x+c) horizontally shifts the graph of f(x) left by c units.

How to Solve Cubic Functions Using the Factor Theorem?

If f(x) = ax ³ + bx ² + cx +d, and f(p) = 0, then x-p is a factor of this cubic function in math.

What is a Rational Function?

A rational function is an algebraic fraction with numerator and denominator as polynomials and the denominator is not equal to zero. The coefficients of the polynomials need not be rational numbers. An example of a rational function: f(x) = 2x/ (x+1)

How to Find the Domain of a Rational Function?

The domain of a rational function consists of all the real numbers x except those for which the denominator is 0. To find these x values to be excluded from the domain of a rational function, equate the denominator to zero and solve for x. For example, f(y) = y/(y ²-25) is a rational function. f(y) = y/[(y+5)(y-5)]. Equating, the denominator to zero, we get the expression to be undefined for y = -5 and y =5. Thus the excluded values are 5, -5.

How do You Prove That a Function is Onto?

A function is onto function if and only if the range is equal to the co-domain.

What is the General Form of a Quadratic Function?

The general form of a quadratic function is f(x)=ax ²+bx+c, where a ≠ 0.

What Are Some Examples of Functions in Math?

The number of sodas coming out of a vending machine depending on how much money you insert.

The amount of carbon left in a fossil after a certain number of years.

The salary of a person is a function of the number of hours worked by that person.

The area of a square is a function of the length of its side. A = a², where a is the length of the side.

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Functions define the relationship between two variables, one is dependent and the other is independent. Function in math is a relation f from a set A (the domain of the function) to another set B (the co-domain of the function). Explore with concept, definition, types, and examples.

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https://www.thoughtco.com/business-math-overview-2312101

Gabinete, Joseph Christian R.

This will help you comprehend on what is business mathematics and teach you about the topics in it.

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https://www.educba.com/what-is-business-mathematics/
This website provides and elaborates on the definition of Business in Mathematics, it's applications in real life and ways on how Mathematics is used in Business.

Rojas, Chris Troy S

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Business Mathematics

Mathematics is an important part of managing business. Business and mathematics go hand in hand this is because business deals with money and money encompasses everything in itself. There is a need for everyone to manage money as some point or the other to take decisions which requires everyone to know mathematics. Business mathematics is used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis. It helps you know the financial formulas, fractions; measurements involved in interest calculation, hire rates, salary calculation, tax calculation etc. which help complete business tasks efficiently. Business mathematics also includes statistics and provides solution to business problems.

Business is always surrounded with challenges which need to be dealt with in a proper fashion so that they do no arise in future. These problems that occur on a daily basis can be effectively solved with the help of mathematical models. Hence mathematics not only helps to calculate but also analyze business problems and work upon them. Learning and using business mathematics enables a person to think out of the box, sharpens one's thinking and helps in precisely formulating and structuring relationships.

Use of Business Mathematics in business

In order to known a business it requires skill more than the developing a product or providing a service. If a business has to survive ad succeed it needs to look after the finances and make necessary arrangements for it to prosper as well. Understanding business mathematics is important to maintain profitable operations and accurate keeping of records. It is required right from the start for pricing products/services till the end when we need to check if the budget was met. Let's look at situations where business mathematics is required:

Before one formally starts production and establishes its business it is very important to estimate the costs that would be incurred in relation to the manufacturing such as the cost of raw materials, machinery, rent, administrative expenses etc. In addition to these basic expenses there are other associated costs such as marketing, warehousing, interest and repayment of loans etc. Once all he expenses relating to production have been included I would be easy to estimate the profit from it to sustain and remain competitive in the market. Accurately determining the cost associated with each item will make the base for the business strong.

When you have successfully determined the costs, the next task is to price the products correctly so that it generates right amount of cash flows for future requirements of the business. Charging the correct selling price would ensure that the product remains profitable.

These require determining the net profit by subtracting the operating costs from the total amount of sales/revenue during a period of time. What also needs to deducted are the tax, depreciation, discount expenses. This helps to find out if the products are being charged enough to continue the business operations and expand.

You need to project the revenue and expenses of a business if we need to analyze the financial health of a business. We need to do sensitivity analysis of how an increase or decrease in sales figure or pricing could affect the business. It helps in determining how each employee contributes to the business and how I would affect. Using business mathematics helps in making these interpretations ad take the business to a higher level.

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Career in Business Mathematics

In addition to mathematical knowledge any person who wants to make their career in business mathematics one needs to also have knowledge of subjects like economics, finance and business management in order to open up many doors in this challenging field. As far the skills goes one has to have good oral and written communication skills as business mathematicians need to work on teams, collate research and be in contact with clients.

Since all corporations require managing their money and business many mathematicians find employment in these fields. Mathematicians follow a logical thinking;follow a problem-solving approach to business.

Companies hire mathematicians to work as the following:

Actuaries

Financial analysts

Insurance underwriters

Budget Analysts

Market Research Analyst

Securities Analyst

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What Is Business Mathematics - Overview of Business Mathematics

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Mathematics is an important part of managing business. Business and mathematics go hand in hand this is because business deals with money and money encompasses everything in itself. There is a need for everyone to manage money as some point or the other to take decisions which requires everyone to know mathematics.

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https://www.vedantu.com/maths/business-mathematics

Loket, Brian Angelo C.

This Website tells us what business mathematics is and the topic.

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Learn about business mathematics topic of Maths in details explained by subject experts on vedantu.com. Register free for online tutoring session to clear your doubts.

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https://www.britannica.com/science/function-mathematics

ModestoFormilleza_this is the link for the definition of functions that will guide you to understand what's in it.

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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https://byjus.com/maths/business-mathematics/
DarleneSerranilla_here you will discover what is business mathematics and what it is all about.

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https://www.thoughtco.com/business-math-overview-2312101

Flores, Elisha D.
Shown here are facts about business math. Which helps us to understand and comprehend with the topic even more.

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What is Business Math? This is the definition of business math, why it is important, and the reasons you should study it.

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Miyoko Miniano
This website shows different content of function.

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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https://byjus.com/maths/business-mathematics/
PhioGodinez_Explains the basics of Business Mathematics and gives basic terms about the topic

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https://www.thoughtco.com/business-math-overview-2312101

Nalaza, John Paul L.

This will guide you to a website that will define what business mathematics is, why you should take it, and teach you the topics of business math.

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What is Business Math? This is the definition of business math, why it is important, and the reasons you should study it.

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This link will take u to a google tab where business mathematics is briefly explained and elaborated

Hayden Villanueva

# data[].attributes.body
https://www.toppr.com/guides/business-mathematics-and-statistics/business-mathematics/laws-of-indices/#Laws_of_Indices

SteffiAnnRosales_ This helps denote the process of taking a power or a root of a number.

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https://www.mathsisfun.com/sets/function.html
CheyneSalvatus_This website will help us understand the definition of functions.

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It is like a machine that has an input and an output.

And the output is related somehow to the input.

f(x)

" f(x) = ... " is the classic way of writing a function.
And there are other ways, as you will see!

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

The input

The relationship

The output

Example: "Multiply by 2" is a very simple function.

Here are the three parts:

For an input of 50, what is the output?

Some Examples of Functions

But we are not going to look at specific functions ...... instead we will look at the
general idea of a function.

Names

First, it is useful to give a function a name.

The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want.

But let's use "f":

We say "f of x equals x squared"

what goes into the function is put inside parentheses () after the name of the function:

And we usually see what a function does with the input:

Example: with f(x) = x²:

In fact we can write f(4) = 16.

The "x" is Just a Place-Holder!

Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

Is the same function as:

The variable (x, q, A, etc) is just there so we know where to put the values:

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

But there is still:

an input (x)

a relationship (squaring)

and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it!

A function relates an input to an output.

Saying " f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:

Here are some example values:

What Types of Things Do Functions Process?

"Numbers" seems an obvious answer, but ...

... which numbers?

For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero.

... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So we need something more powerful, and that is where sets come in:

A set is a collection of things.

Here are some examples:

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

Set of clothes: {"hat","shirt",...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}

Each individual thing in the set (such as "4" or "hat") is called a member, or element.

So, a function takes elements of a set, and gives back elements of a set.

A Function is Special

But a function has special rules:

It must work for every possible input value

And it has only one relationship for each input value

This can be said in one definition:

Formal Definition of a Function

A function relates each element of a set
with exactly one element of another set
(possibly the same set).

The Two Important Things!

"...each element..." means that every element in X is related to some element in Y.

We say that the function covers X (relates every element of it).

(But some elements of Y might not be related to at all, which is fine.)

When a relationship does not follow those two rules then it is not a function ... it is still a relationship, just not a function.

Example: The relationship x → x²

Could also be written as a table:

It is a function, because:

Every element in X is related to Y

No element in X has two or more relationships

So it follows the rules.

(Notice how both 4 and -4 relate to 16, which is allowed.)

Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:

Value "3" in X has no relation in Y

Value "4" in X has no relation in Y

Value "5" is related to more than one value in Y

(But the fact that "6" in Y has no relationship does not matter)

Vertical Line Test

On a graph, the idea of single valued means that no vertical line ever crosses more than one value.

If it crosses more than once it is still a valid curve, but is not a function.

Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective

Infinitely Many

My examples have just a few values, but functions usually work on sets with infinitely many elements.

Example: y = x³

The input set "X" is all Real Numbers

The output set "Y" is also all the Real Numbers

We can't show ALL the values, so here are just a few examples:

Domain, Codomain and Range

In our examples above

the set "X" is called the Domain,

the set "Y" is called the Codomain, and

the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

We have a special page on Domain, Range and Codomain if you want to know more.

So Many Names!

Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.

Here are some common terms you should get familiar with:

Example: z = 2u³:

"u" could be called the "independent variable"

"z" could be called the "dependent variable" (it depends on the value of u)

Example: f(4) = 16:

"4" could be called the "argument"

"16" could be called the "value of the function"

Example: h(year) = 20 × year:

h() is the function

"year" could be called the "argument", or the "variable"

For software "year" is called the parameter, and the value given to the parameter is called the variable!
We often call a function "f(x)" when in fact the function is really "f"

Ordered Pairs

And here is another way to think about functions:

Write the input and output of a function as an "ordered pair", such as (4,16).

They are called ordered pairs because the input always comes first, and the output second:

(input, output)

So it looks like this:

Example:

(4,16) means that the function takes in "4" and gives out "16"

Set of Ordered Pairs

A function can then be defined as a set of ordered pairs:

Example: {(2,4), (3,5), (7,3)} is a function that says

"2 is related to 4", "3 is related to 5" and "7 is related 3".

Also, notice that:

the domain is {2,3,7} (the input values)

and the range is {4,5,3} (the output values)

But the function has to be single valued, so we also say

"if it contains (a, b) and (a, c), then b must equal c"

Which is just a way of saying that an input of "a" cannot produce two different results.

Example: {( 2, 4), ( 2, 5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5.

In other words it is not a function because it is not single valued

A Benefit of Ordered Pairs

We can graph them...

... because they are also coordinates!

So a set of coordinates is also a function (if they follow the rules above, that is)

A Function Can be in Pieces

We can create functions that behave differently depending on the input value

Example: A function with two pieces:

when x is less than 0, it gives 5,

when x is 0 or more it gives x²

Here are some example values:

x

y

-3

5

-1

5

0

0

2

4

4

16

...

...

Read more at Piecewise Functions.

Explicit vs Implicit

One last topic: the terms "explicit" and "implicit".

Explicit is when the function shows us how to go directly from x to y, such as:
When we know x, we can find y
That is the classic y = f(x) style that we often work with.

Implicit is when it is not given directly such as:
When we know x, how do we find y?
It may be hard (or impossible!) to go directly from x to y.

"Implicit" comes from "implied", in other words shown indirectly.

Graphing

Conclusion

a function relates inputs to outputs

a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).

all the outputs (the actual values related to) are together called the range

a function is a special type of relation where:

every element in the domain is included, and

any input produces only one output (not this or that)

an input and its matching output are together called an ordered pair

so a function can also be seen as a set of ordered pairs

5571, 5572, 535, 5207, 5301, 1173, 7281, 533, 8414, 8430

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It is like a machine that has an input and an output. And the output is related somehow to the input. f(x) " f(x) = ... " is the classic way of writing a function. And there are other ways, as you will see!

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Rhyme Godfrey Anque_this link will help you to understand the functions, definition, types, examples and facts of mathematics

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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https://byjus.com/maths/business-mathematics/

The link that is attached here can give us insight into the basic information about the topic of Business Mathematics, such as definitions and basic terms.

Pantas, Aicelle P.

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https://www.google.com/amp/s/www.geeksforgeeks.org/proposition-logic/amp/

This website provides the definition of Logic in Mathematics, it also shows prepositional logic and other important topics such as The Truth Table.

Rojas, Chris Troy S

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https://mathvault.ca/hub/higher-math/math-symbols/logic-symbols/

Gabinete, Joseph Christian R.

This will introduce to you on what logic
symbols is and types of logic symbols

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This website will take u to a tab where Logic and paradoxes is brieflyy explained and elaborated

Hayden Villanueva

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Logic and Paradoxes | World of Mathematics – Mathigon

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Introduction – Propositional Calculus – Logic Tables | An interactive textbook

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https://mathigon.org/world/Logic_and_Paradoxes

Loket, Brian Angelo C.

This website shows the introduction about logic

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Logic and Paradoxes | World of Mathematics – Mathigon

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Introduction – Propositional Calculus – Logic Tables | An interactive textbook

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https://www.vedantu.com/maths/business-mathematics

Dragon, Jandex Nicole P.
This is to give you insights about what Business Mathematics is and its scopes and more importantly, what it covers.

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Learn about business mathematics topic of Maths in details explained by subject experts on vedantu.com. Register free for online tutoring session to clear your doubts.

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https://www.vedantu.com/maths/business-mathematics

ModestoFormilleza_ This link helps you to learn and guide you to know more about business mathematics.

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Learn about business mathematics topic of Maths in details explained by subject experts on vedantu.com. Register free for online tutoring session to clear your doubts.

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Business Mathematics | Formulas and etc.

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Business Mathematics | Formulas and etc.

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https://www.dummies.com/article/business-careers-money/business/general-business/business-math-for-dummies-cheat-sheet-208644/

dela Rosa, Rob Aileron R.
This shows you the several formulas that is used in general mathematics as well as common acronyms, interest rates, equivalents, etc.

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Math is an important part of managing business. Get to know some commonly used fractions and their decimal equivalents, area and perimeter formulas, angle measurements, and financial formulas - including understanding interest rates and common financial acronyms - to help with your business tasks.

Earning money through interest

About This Article

This article is from the book:

About the book authors:

Mary Jane Sterling is the author of four other For Dummies titles: Algebra For Dummies, Algebra II For Dummies, Trigonometry For Dummies, and Math Word Problems For Dummies. She has honed her math-explaining skills during her years of teaching mathematics at all levels: junior high school, high school, and college. She has been teaching at Bradley University, in Peoria, Illinois, for almost 30 of those years.
When not teaching or writing, Mary Jane keeps busy by working with her Kiwanis Club, advising Bradley University's Circle K Club, and working with members of the Heart of Illinois Aktion Club (for adults with disabilities). All the volunteer projects taken on for these clubs help keep her busy and involved in the community.

This article can be found in the category:

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Business Math For Dummies Cheat Sheet - dummies

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Math is an important part of managing business. Get to know some commonly used fractions and their decimal equivalents, area and perimeter formulas, angle measurements, and financial formulas - including understanding interest rates and common financial acronyms - to help with your business tasks.

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https://byjus.com/maths/mathematical-logic/
DarleneSerranilla_all the things you should know about logic are indicated in this link and this will serve as your guide throughout the lesson.

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Mathematical Logic (AND, OR & NOT) | Formulas and Examples

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The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

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This website can help us understand business mathematics. This website gives a lot of information. This website will help you prepare for business mathematics.

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What is Business Math? This is the definition of business math, why it is important, and the reasons you should study it.

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https://faculty.math.illinois.edu/~vddries/main2.pdf

SarahBelleza_This is a pdf file that will introduce to logic in mathematics and talk about all the other different topics of logic

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A function is defined as a relation between a set of inputs having one output each and the function is generally denoted by f(x) where "x" is the input, the general representation of a function is y = f(x).
https://byjus.com/maths/what-is-a-function/ Christian Paul A. Torrico_ Key Takeaway

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What is a Function? Definition, Types and Notation

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Learn what is a function along with its types and examples. A function in Maths is defined as a relation between two or more variables, whose graph can be plotted in cartesian plane.

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https://mathigon.org/world/Logic_and_Paradoxes

Nalaza, John Paul L.

This will lead you to an introduction of what is Logic and how is it used in mathematics.

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Rhyme Godfrey Anque_This link will send you to know what is the real definition of math logic and it explains the true concept of logic itself

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Mathematical Logic (AND, OR & NOT) | Formulas and Examples

# data[].attributes.attachment_link.description
The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

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https://byjus.com/maths/mathematical-logic/

Flores, Elisha D.
The link given enables us to be familiar and learn more about logic in mathematics. And tells us more about the given topic.

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Mathematical Logic (AND, OR & NOT) | Formulas and Examples

# data[].attributes.attachment_link.description
The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

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https://mathigon.org/world/Logic_and_Paradoxes

ModestoFormilleza_this link will let you know what is true or false, right or wrongm and etc. It can be use in our everyday lives.

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Logic and Paradoxes | World of Mathematics – Mathigon

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Introduction – Propositional Calculus – Logic Tables | An interactive textbook

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https://www.youtube.com/watch?v=kHmFDkQKKMI

Dragon, Jandex Nicole P.
A six minute video showing what Logic in Grade 11 Mathematics is and a further look on what this topic will be solving

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https://www.britannica.com/science/function-mathematics

On this website, you will be provided with informations about Functions and help us understand this topic.

Domingo, Ma. Paulina Beatriz D.

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function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

This relationship is commonly symbolized as y = f( x)-which is said " f of x"-and y and x are related such that for every x, there is a unique value of y. That is, f( x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f( x) in another set. The set of values of x is called the domain of the function, and the set of values of f( x) generated by the values in the domain is called the range of the function. In addition to f( x), other abbreviated symbols such as g( x) and P( x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified.

Common functions

Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = π r ², gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = b h/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values-thus, any real number -the functions are known as real-valued functions.

The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P( x) = a ₀ + a ₁ x + a ₂ x ²+⋯+ a n x n, where the coefficients ( a ₀, a ₁, a ₂,..., a n) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,...). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility-practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five-linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x- y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f( x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates ( x, y) where y = f( x). For example, the graph of the cubic equation f( x) = x ³ − 3 x + 2 is shown in the figure.

Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle ( see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or "cycles."

The exponential function is a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = e x, sometimes written y = exp ( x), in which e (2.7182818...) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = e x, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.

Complex functions

Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + i y, where i is the imaginary unit (the square root of −1) and x and y are real variables ( see figure), it is possible to split the complex function into real and imaginary parts: f( z) = P( x, y) + i Q( x, y).

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Function | Definition, Types, Examples, & Facts

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Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.

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https://plato.stanford.edu/entries/model-theory/

SteffiAnnRosales_ It is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures.

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Sometimes we write or speak a sentence \(S\) that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that \(S\) comes to express a true or false statement, we are said to \(S\), and the added information is called an interpretation of \(S\). If the interpretation \(I\) happens to make \(S\) state something true, we say that \(I\) is a model of \(S\), or that \(I\) satisfies \(S\), in symbols '\(I \vDash S\)'. Another way of saying that \(I\) is a model of \(S\) is to say that \(S\) is true in \(I\), and so we have the notion of model-theoretic truth, which is truth in a particular interpretation. But one should remember that the statement '\(S\) is true in \(I\)' is just a paraphrase of '\(S\), when interpreted as in \(I\), is true'; so model-theoretic truth is parasitic on plain ordinary truth, and we can always paraphrase it away.

For example I might say

He is killing all of them,

and offer the interpretation that 'he' is Alfonso Arblaster of 35 The Crescent, Beetleford, and that 'them' are the pigeons in his loft. This interpretation explains (a) what objects some expressions refer to, and (b) what classes some quantifiers range over. (In this example there is one quantifier: 'all of them'). Interpretations that consist of items (a) and (b) appear very often in model theory, and they are known as structures. Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use so-called first-order structures, model theory of modal logics uses Kripke structures, and so on.

The structure \(I\) in the previous paragraph involves one fixed object and one fixed class. Since we described the structure today, the class is the class of pigeons in Alfonso's loft today, not those that will come tomorrow to replace them. If Alfonso Arblaster kills all the pigeons in his loft today, then \(I\) satisfies the quoted sentence today but won't satisfy it tomorrow, because Alfonso can't kill the same pigeons twice over. Depending on what you want to use model theory for, you may be happy to evaluate sentences today (the default time), or you may want to record how they are satisfied at one time and not at another. In the latter case you can relativise the notion of model and write '\(I \vDash_t S\)' to mean that \(I\) is a model of \(S\) at time \(t\). The same applies to places, or to anything else that might be picked up by other implicit indexical features in the sentence. For example if you believe in possible worlds, you can index \(\vDash\) by the possible world where the sentence is to be evaluated. Apart from using set theory, model theory is completely agnostic about what kinds of thing exist.

Note that the objects and classes in a structure carry labels that steer them to the right expressions in the sentence. These labels are an essential part of the structure.

If the same class is used to interpret all quantifiers, the class is called the domain or universe of the structure. But sometimes there are quantifiers ranging over different classes. For example if I say

One of those thingummy diseases is killing all the birds.

you will look for an interpretation that assigns a class of diseases to 'those thingummy diseases' and a class of birds to 'the birds'. Interpretations that give two or more classes for different quantifiers to range over are said to be many-sorted, and the classes are sometimes called the sorts.

The ideas above can still be useful if we start with a sentence \(S\) that does say something either true or false without needing further interpretation. (Model theorists say that such a sentence is fully interpreted.) For example we can consider misinterpretations \(I\) of a fully interpreted sentence \(S\). A misinterpretation of \(S\) that makes it true is known as a nonstandard or unintended model of \(S\). The branch of mathematics called nonstandard analysis is based on nonstandard models of mathematical statements about the real or complex number systems; see Section 4 below.

One also talks of model-theoretic semantics of natural languages, which is a way of describing the meanings of natural language sentences, not a way of giving them meanings. The connection between this semantics and model theory is a little indirect. It lies in Tarski's truth definition of 1933. See the entry on Tarski's truth definitions for more details.

A sentence \(S\) divides all its possible interpretations into two classes, those that are models of it and those that are not. In this way it defines a class, namely the class of all its models, written \(\Mod(S)\). To take a legal example, the sentence

The first person has transferred the property to the second person, who thereby holds the property for the benefit of the third person.

defines a class of structures which take the form of labelled 4-tuples, as for example (writing the label on the left):

the first person = Alfonso Arblaster;

the property = the derelict land behind Alfonso's house;

the second person = John Doe;

the third person = Richard Roe.

This is a typical model-theoretic definition, defining a class of structures (in this case, the class known to the lawyers as trusts).

We can extend the idea of model-theoretic definition from a single sentence \(S\) to a set \(T\) of sentences; \(\Mod(T)\) is the class of all interpretations that are simultaneously models of all the sentences in \(T\). When a set \(T\) of sentences is used to define a class in this way, mathematicians say that \(T\) is a theory or a set of axioms, and that \(T\) axiomatises the class \(\Mod(T)\).

Take for example the following set of first-order sentences:

\[\begin{align*} & \forall x\forall y\forall z (x + (y + z) = (x + y) + z). \\ & \forall x (x + 0 = x). \\ & \forall x (x + (-x) = 0). \\ & \forall x\forall y (x + y = y + x). \end{align*}\]

Here the labels are the addition symbol '+', the minus symbol '\(-\)' and the constant symbol '0'. An interpretation also needs to specify a domain for the quantifiers. With one proviso, the models of this set of sentences are precisely the structures that mathematicians know as abelian groups. The proviso is that in an abelian group \(A\), the domain should contain the interpretation of the symbol 0, and it should be closed under the interpretations of the symbols + and \(-\). In mathematical model theory one builds this condition (or the corresponding conditions for other function and constant symbols) into the definition of a structure.

Each mathematical structure is tied to a particular first-order language. A structure contains interpretations of certain predicate, function and constant symbols; each predicate or function symbol has a fixed arity. The collection \(K\) of these symbols is called the signature of the structure. Symbols in the signature are often called nonlogical constants, and an older name for them is primitives. The first-order language of signature \(K\) is the first-order language built up using the symbols in \(K\), together with the equality sign =, to build up its atomic formulas. (See the entry on classical logic.) If \(K\) is a signature, \(S\) is a sentence of the language of signature \(K\) and \(A\) is a structure whose signature is \(K\), then because the symbols match up, we know that \(A\) makes \(S\) either true or false. So one defines the class of abelian groups to be the class of all those structures of signature \(+\), \(-\), \(0\) which are models of the sentences above. Apart from the fact that it uses a formal first-order language, this is exactly the algebraists' usual definition of the class of abelian groups; model theory formalises a kind of definition that is extremely common in mathematics.

Now the defining axioms for abelian groups have three kinds of symbol (apart from punctuation). First there is the logical symbol = with a fixed meaning. Second there are the nonlogical constants, which get their interpretation by being applied to a particular structure; one should group the quantifier symbols with them, because the structure also determines the domain over which the quantifiers range. And third there are the variables \(x, y\) etc. This three-level pattern of symbols allows us to define classes in a second way. Instead of looking for the interpretations of the nonlogical constants that will make a sentence true, we fix the interpretations of the nonlogical constants by choosing a particular structure \(A\), and we look for assignments of elements of \(A\) to variables which will make a given formula true in \(A\).

For example let \(\mathbb{Z}\) be the additive group of integers. Its elements are the integers (positive, negative and 0), and the symbols \(+\), \(-\), \(0\) have their usual meanings. Consider the formula

\[ v_1 + v_1 = v_2. \]

If we assign the number \(-3\) to \(v_1\) and the number \(-6\) to \(v_2\), the formula works out as true in \(\mathbb{Z}\). We express this by saying that the pair \((-3,-6)\) satisfies this formula in \(\mathbf{Z}\). Likewise (15,30) and (0,0) satisfy it, but \((2,-4)\) and (3,3) don't. Thus the formula defines a binary relation on the integers, namely the set of pairs of integers that satisfy it. A relation defined in this way in a structure \(A\) is called a first-order definable relation in \(A\). A useful generalisation is to allow the defining formula to use added names for some specific elements of \(A\); these elements are called parameters and the relation is then definable with parameters.

This second type of definition, defining relations inside a structure rather than classes of structure, also formalises a common mathematical practice. But this time the practice belongs to geometry rather than to algebra. You may recognise the relation in the field of real numbers defined by the formula

\[ v_1^2 + v_2^2 = 1. \]

It's the circle of radius 1 around the origin in the real plane. Algebraic geometry is full of definitions of this kind.

During the 1940s it occurred to several people (chiefly Anatolii Mal'tsev in Russia, Alfred Tarski in the USA and Abraham Robinson in Britain) that the metatheorems of classical logic could be used to prove mathematical theorems about classes defined in the two ways we have just described. In 1950 both Robinson and Tarski were invited to address the International Congress of Mathematicians at Cambridge Mass. on this new discipline (which as yet had no name - Tarski proposed the name 'theory of models' in 1954). The conclusion of Robinson's address to that Congress is worth quoting:

[The] concrete examples produced in the present paper will have shown that contemporary symbolic logic can produce useful tools - though by no means omnipotent ones - for the development of actual mathematics, more particularly for the development of algebra and, it would appear, of algebraic geometry. This is the realisation of an ambition which was expressed by Leibniz in a letter to Huyghens as long ago as 1679. (Robinson 1952, 694)

In fact Mal'tsev had already made quite deep applications of model theory in group theory several years earlier, but under the political conditions of the time his work in Russia was not yet known in the West. By the end of the twentieth century, Robinson's hopes had been amply fulfilled; see the entry on first-order model theory.

There are at least two other kinds of definition in model theory besides these two above. The third is known as interpretation (a special case of the interpretations that we began with). Here we start with a structure \(A\), and we build another structure \(B\) whose signature need not be related to that of \(A\), by defining the domain \(X\) of \(B\) and all the labelled relations and functions of \(B\) to be the relations definable in \(A\) by certain formulas with parameters. A further refinement is to find a definable equivalence relation on \(X\) and take the domain of \(B\) to be not \(X\) itself but the set of equivalence classes of this relation. The structure \(B\) built in this way is said to be interpreted in the structure \(A\).

A simple example, again from standard mathematics, is the interpretation of the group \(\mathbb{Z}\) of integers in the structure \(\mathbb{N}\) consisting of the natural numbers 0, 1, 2 etc. with labels for 0, 1 and +. To construct the domain of \(\mathbb{Z}\) we first take the set \(X\) of all ordered pairs of natural numbers (clearly a definable relation in \(\mathbb{N})\), and on this set \(X\) we define the equivalence relation \(\sim\) by

\[ (a,b) \sim(c,d) \text{ if and only if } a + d = b + c \]

(again definable). The domain of \(\mathbb{Z}\) consists of the equivalence classes of this relation. We define addition on \(\mathbb{Z}\) by

\[ (a,b) + (c,d) = (e,f) \text{ if and only if } a + c + f = b + d + e. \]

The equivalence class of \((a,b)\) becomes the integer \(a - b\).

When a structure \(B\) is interpreted in a structure \(A\), every first-order statement about \(B\) can be translated back into a first-order statement about \(A\), and in this way we can read off the complete theory of \(B\) from that of \(A\). In fact if we carry out this construction not just for a single structure \(A\) but for a family of models of a theory \(T\), always using the same defining formulas, then the resulting structures will all be models of a theory \(T'\) that can be read off from \(T\) and the defining formulas. This gives a precise sense to the statement that the theory \(T'\) is interpreted in the theory \(T\). Philosophers of science have sometimes experimented with this notion of interpretation as a way of making precise what it means for one theory to be reducible to another. But realistic examples of reductions between scientific theories seem generally to be much subtler than this simple-minded model-theoretic idea will allow. See the entry on intertheory relations in physics.

The fourth kind of definability is a pair of notions, implicit definability and explicit definability of a particular relation in a theory. See section 3.3 of the entry on first-order model theory.

Unfortunately there used to be a very confused theory about model-theoretic axioms, that also went under the name of implicit definition. By the end of the nineteenth century, mathematical geometry had generally ceased to be a study of space, and it had become the study of classes of structures which satisfy certain 'geometric' axioms. Geometric terms like 'point', 'line' and 'between' survived, but only as the primitive symbols in axioms; they no longer had any meaning associated with them. So the old question, whether Euclid's parallel postulate (as a statement about space) was deducible from Euclid's other assumptions about space, was no longer interesting to geometers. Instead, geometers showed that if one wrote down an up-to-date version of Euclid's other assumptions, in the form of a theory \(T\), then it was possible to find models of \(T\) which fail to satisfy the parallel postulate. (See the entry on geometry in the 19th century for the contributions of Lobachevski and Klein to this achievement.) In 1899 David Hilbert published a book in which he constructed such models, using exactly the method of interpretation that we have just described.

Problems arose because of the way that Hilbert and others described what they were doing. The history is complicated, but roughly the following happened. Around the middle of the nineteenth century people noticed, for example, that in an abelian group the minus function is definable in terms of 0 and + (namely: \(-a\) is the element \(b\) such that \(a + b = 0)\). Since this description of minus is in fact one of the axioms defining abelian groups, we can say (using a term taken from J. D. Gergonne, who should not be held responsible for the later use made of it) that the axioms for abelian groups implicitly define minus. In the jargon of the time, one said not that the axioms define the function minus, but that they define the concept minus. Now suppose we switch around and try to define plus in terms of minus and 0. This way round it can't be done, since one can have two abelian groups with the same 0 and minus but different plus functions. Rather than say this, the nineteenth century mathematicians concluded that the axioms only partially define plus in terms of minus and 0. Having swallowed that much, they went on to say that the axioms together form an implicit definition of the concepts plus, minus and 0 together, and that this implicit definition is only partial but it says about these concepts precisely as much as we need to know.

One wonders how it could happen that for fifty years nobody challenged this nonsense. In fact some people did challenge it, notably the geometer Moritz Pasch who in section 12 of his Vorlesungen über Neuere Geometrie (1882) insisted that geometric axioms tell us nothing whatever about the meanings of 'point', 'line' etc. Instead, he said, the axioms give us relations between the concepts. If one thinks of a structure as a kind of ordered \(n\)-tuple of sets etc., then a class \(\Mod(T)\) becomes an \(n\)-ary relation, and Pasch's account agrees with ours. But he was unable to spell out the details, and there is some evidence that his contemporaries (and some more recent commentators) thought he was saying that the axioms may not determine the meanings of 'point' and 'line', but they do determine those of relational terms such as 'between' and 'incident with'! Frege's demolition of the implicit definition doctrine was masterly, but it came too late to save Hilbert from saying, at the beginning of his Grundlagen der Geometrie, that his axioms give 'the exact and mathematically adequate description' of the relations 'lie', 'between' and 'congruent'. Fortunately Hilbert's mathematics speaks for itself, and one can simply bypass these philosophical faux pas. The model-theoretic account that we now take as a correct description of this line of work seems to have surfaced first in the group around Giuseppe Peano in the 1890s, and it reached the English-speaking world through Bertrand Russell's Principles of Mathematics in 1903.

Suppose \(L\) is a language of signature \(K, T\) is a set of sentences of \(L\) and \(\phi\) is a sentence of \(L\). Then the relation

\[ \Mod(T) \subseteq \Mod(\phi) \]

expresses that every structure of signature \(K\) which is a model of \(T\) is also a model of \(\phi\). This is known as the model-theoretic consequence relation, and it is written for short as

\[ T \vDash \phi \]

The double use of \(\vDash\) is a misfortune. But in the particular case where \(L\) is first-order, the completeness theorem (see the entry on classical logic) tells us that '\(T \vDash \phi\)' holds if and only if there is a proof of \(\phi\) from \(T\), a relation commonly written

\[ T \vdash \phi \]

Since \(\vDash\) and \(\vdash\) express exactly the same relation in this case, model theorists often avoid the double use of \(\vDash\) by using \(\vdash\) for model-theoretic consequence. But since what follows is not confined to first-order languages, safety suggests we stick with \(\vDash\) here.

Before the middle of the nineteenth century, textbooks of logic commonly taught the student how to check the validity of an argument (say in English) by showing that it has one of a number of standard forms, or by paraphrasing it into such a form. The standard forms were syntactic and/or semantic forms of argument in English. The process was hazardous: semantic forms are almost by definition not visible on the surface, and there is no purely syntactic form that guarantees validity of an argument. For this reason most of the old textbooks had a long section on 'fallacies' - ways in which an invalid argument may seem to be valid.

In 1847 George Boole changed this arrangement. For example, to validate the argument

All monarchs are human beings. No human beings are infallible. Therefore no infallible beings are monarchs.

Boole would interpret the symbols \(P, Q, R\) as names of classes:

\(P\) is the class of all monarchs.
\(Q\) is the class of all human beings.
\(R\) is the class of all infallible beings.

Then he would point out that the original argument paraphrases into a set-theoretic consequence:

\[ (P \subseteq Q), (Q \cap R = 0) \vDash(R \cap P = 0) \]

(This example is from Stanley Jevons, 1869. Boole's own account is idiosyncratic, but I believe Jevons' example represents Boole's intentions accurately.) Today we would write \(\forall x(Px \rightarrow Qx)\) rather than \(P \subseteq Q\), but this is essentially the standard definition of \(P \subseteq Q\), so the difference between us and Boole is slight.

Insofar as they follow Boole, modern textbooks of logic establish that English arguments are valid by reducing them to model-theoretic consequences. Since the class of model-theoretic consequences, at least in first-order logic, has none of the vaguenesses of the old argument forms, textbooks of logic in this style have long since ceased to have a chapter on fallacies.

But there is one warning that survives from the old textbooks: If you formalise your argument in a way that is not a model-theoretic consequence, it doesn't mean the argument is not valid. It may only mean that you failed to analyse the concepts in the argument deeply enough before you formalised. The old textbooks used to discuss this in a ragbag section called 'topics' (i.e. hints for finding arguments that you might have missed). Here is an example from Peter of Spain's 13th century Summulae Logicales:

'There is a father. Therefore there is a child.' ... Where does the validity of this argument come from? From the relation. The maxim is: When one of a correlated pair is posited, then so is the other.

Hilbert and Ackermann, possibly the textbook that did most to establish the modern style, discuss in their section III.3 a very similar example: 'If there is a son, then there is a father'. They point out that any attempt to justify this by using the symbolism

\[ \exists xSx \rightarrow \exists xFx \]

is doomed to failure. "A proof of this statement is possible only if we analyze conceptually the meanings of the two predicates which occur", as they go on to illustrate. And of course the analysis finds precisely the relation that Peter of Spain referred to.

On the other hand if your English argument translates into an invalid model-theoretic consequence, a counterexample to the consequence may well give clues about how you can describe a situation that would make the premises of your argument true and the conclusion false. But this is not guaranteed.

One can raise a number of questions about whether the modern textbook procedure does really capture a sensible notion of logical consequence. For example in Boole's case the set-theoretic consequences that he relies on are all easily provable by formal proofs in first-order logic, not even using any set-theoretic axioms; and by the completeness theorem (see the entry on classical logic) the same is true for first-order logic. But for some other logics it is certainly not true. For instance the model-theoretic consequence relation for some logics of time presupposes some facts about the physical structure of time. Also, as Boole himself pointed out, his translation from an English argument to its set-theoretic form requires us to believe that for every property used in the argument, there is a corresponding class of all the things that have the property. This comes dangerously close to Frege's inconsistent comprehension axiom!

In 1936 Alfred Tarski proposed a definition of logical consequence for arguments in a fully interpreted formal language. His proposal was that an argument is valid if and only if: under any allowed reinterpretation of its nonlogical symbols, if the premises are true then so is the conclusion. Tarski assumed that the class of allowed reinterpretations could be read off from the semantics of the language, as set out in his truth definition. He left it undetermined what symbols count as nonlogical; in fact he hoped that this freedom would allow one to define different kinds of necessity, perhaps separating 'logical' from 'analytic'. One thing that makes Tarski's proposal difficult to evaluate is that he completely ignores the question we discussed above, of analysing the concepts to reach all the logical connections between them. The only plausible explanation I can see for this lies in his parenthetical remark about

the necessity of eliminating any defined signs which may possibly occur in the sentences concerned, i.e. of replacing them by primitive signs.

This suggests to me that he wants his primitive signs to be by stipulation unanalysable. But then by stipulation it will be purely accidental if his notion of logical consequence captures everything one would normally count as a logical consequence.

Historians note a resemblance between Tarski's proposal and one in section 147 of Bernard Bolzano's Wissenschaftslehre of 1837. Like Tarski, Bolzano defines the validity of a proposition in terms of the truth of a family of related propositions. Unlike Tarski, Bolzano makes his proposal for propositions in the vernacular, not for sentences of a formal language with a precisely defined semantics.

On all of this section, see also the entry on logical consequence.

A sentence \(S\) defines its class \(\Mod(S)\) of models. Given two languages \(L\) and \(L'\), we can compare them by asking whether every class \(\Mod(S)\), with \(S\) a sentence of \(L\), is also a class of the form \(\Mod(S')\) where \(S'\) is a sentence of \(L'\). If the answer is Yes, we say that \(L\) is reducible to \(L'\), or that \(L'\) is at least as expressive as \(L\).

For example if \(L\) is a first-order language with identity, whose signature consists of 1-ary predicate symbols, and \(L'\) is the language whose sentences consist of the four syllogistic forms (All \(A\) are \(B\), Some \(A\) are \(B\), No \(A\) are \(B\), Some \(A\) are not \(B)\) using the same predicate symbols, then \(L'\) is reducible to \(L\), because the syllogistic forms are expressible in first-order logic. (There are some quarrels about which is the right way to express them; see the entry on the traditional square of opposition.) But the first-order language \(L\) is certainly not reducible to the language \(L'\) of syllogisms, since in \(L\) we can write down a sentence saying that exactly three elements satisfy \(Px\), and there is no way of saying this using just the syllogistic forms. Or moving the other way, if we form a third language \(L''\) by adding to \(L\) the quantifier \(Qx\) with the meaning "There are uncountably many elements \(x\) such that ...", then trivially \(L\) is reducible to \(L''\), but the downward Loewenheim-Skolem theorem shows at once that \(L''\) is not reducible to \(L\).

These notions are useful for analysing the strength of database query languages. We can think of the possible states of a database as structures, and a simple Yes/No query becomes a sentence that elicits the answer Yes if the database is a model of it and No otherwise. If one database query language is not reducible to another, then the first can express some query that can't be expressed in the second.

So we need techniques for comparing the expressive strengths of languages. One of the most powerful techniques available consists of the back-and-forth games of Ehrenfeucht and Fraïssé between the two players Spoiler and Duplicator; see the entry on logic and games for details. Imagine for example that we play the usual first-order back-and-forth game \(G\) between two structures \(A\) and \(B\). The theory of these games establishes that if some first-order sentence \(\phi\) is true in exactly one of \(A\) and \(B\), then there is a number \(n\), calculable from \(\phi\), with the property that Spoiler has a strategy for \(G\) that will guarantee that he wins in at most \(n\) steps. So conversely, to show that first-order logic can't distinguish between \(A\) and \(B\), it suffices to show that for every finite \(n\), Duplicator has a strategy that will guarantee she doesn't lose \(G\) in the first \(n\) steps. If we succeed in showing this, it follows that any language which does distinguish between \(A\) and \(B\) is not reducible to the first-order language of the structures \(A\) and \(B\).

These back-and-forth games are immensely flexible. For a start, they make just as much sense on finite structures as they do on infinite; many other techniques of classical model theory assume that the structures are infinite. They can also be adapted smoothly to many non-first-order languages.

In 1969 Per Lindström used back-and-forth games to give some abstract characterisations of first-order logic in terms of its expressive power. One of his theorems says that if \(L\) is a language with a signature \(K, L\) is closed under all the first-order syntactic operations, and \(L\) obeys the downward Loewenheim-Skolem theorem for single sentences, and the compactness theorem, then \(L\) is reducible to the first-order language of signature \(K\). These theorems are very attractive; see Chapter XII of Ebbinghaus, Flum and Thomas for a good account. But they have never quite lived up to their promise. It has been hard to find any similar characterisations of other logics. Even for first-order logic it is a little hard to see exactly what the characterisations tell us. But very roughly speaking, they tell us that first-order logic is the unique logic with two properties: (1) we can use it to express arbitrarily complicated things about finite patterns, and (2) it is hopeless for discriminating between one infinite cardinal and another.

These two properties (1) and (2) are just the properties of first-order logic that allowed Abraham Robinson to build his nonstandard analysis. The background is that Leibniz, when he invented differential and integral calculus, used infinitesimals, i.e. numbers that are greater than 0 and smaller than all of 1/2, 1/3, 1/4 etc. Unfortunately there are no such real numbers. During the nineteenth century all definitions and proofs in the Leibniz style were rewritten to talk of limits instead of infinitesimals. Now let \(\mathbb{R}\) be the structure consisting of the field of real numbers together with any structural features we care to give names to: certainly plus and times, maybe the ordering, the set of integers, the functions sin and log, etc. Let \(L\) be the first-order language whose signature is that of \(\mathbb{R}\). Because of the expressive strength of \(L\), we can write down any number of theorems of calculus as sentences of \(L\). Because of the expressive weakness of \(L\), there is no way that we can express in \(L\) that \(\mathbb{R}\) has no infinitesimals. In fact Robinson used the compactness theorem to build a structure \(\mathbb{R}'\) that is a model of exactly the same sentences of \(L\) as \(\mathbb{R}\), but which has infinitesimals. As Robinson showed, we can copy Leibniz's arguments using the infinitesimals in \(\mathbb{R}'\), and so prove that various theorems of calculus are true in \(\mathbb{R}'\). But these theorems are expressible in \(L\), so they must also be true in \(\mathbb{R}\).

Since arguments using infinitesimals are usually easier to visualise than arguments using limits, nonstandard analysis is a helpful tool for mathematical analysts. Jacques Fleuriot in his Ph.D. thesis (2001) automated the proof theory of nonstandard analysis and used it to mechanise some of the proofs in Newton's Principia.

To model a phenomenon is to construct a formal theory that describes and explains it. In a closely related sense, you model a system or structure that you plan to build, by writing a description of it. These are very different senses of 'model' from that in model theory: the 'model' of the phenomenon or the system is not a structure but a theory, often in a formal language. The Universal Modeling Language, UML for short, is a formal language designed for just this purpose. It's reported that the Australian Navy once hired a model theorist for a job 'modelling hydrodynamic phenomena'. (Please don't enlighten them!)

A little history will show how the word 'model' came to have these two different uses. In late Latin a 'modellus' was a measuring device, for example to measure water or milk. By the vagaries of language, the word generated three different words in English: mould, module, model. Often a device that measures out a quantity of a substance also imposes a form on the substance. We see this with a cheese mould, and also with the metal letters (called 'moduli' in the early 17th century) that carry ink to paper in printing. So 'model' comes to mean an object in hand that expresses the design of some other objects in the world: the artist's model carries the form that the artist depicts, and Christopher Wren's 'module' of St Paul's Cathedral serves to guide the builders.

Already by the late 17th century the word 'model' could mean an object that shows the form, not of real-world objects, but of mathematical constructs. Leibniz boasted that he didn't need models in order to do mathematics. Other mathematicians were happy to use plaster or metal models of interesting surfaces. The models of model theory first appeared as abstract versions of this kind of model, with theories in place of the defining equation of a surface. On the other hand one could stay with real-world objects but show their form through a theory rather than a physical copy in hand; 'modelling' is building such a theory.

We have a confusing halfway situation when a scientist describes a phenomenon in the world by an equation, for example a differential equation with exponential functions as solutions. Is the model the theory consisting of the equation, or are these exponential functions themselves models of the phenomenon? Examples of this kind, where theory and structures give essentially the same information, provide some support for Patrick Suppes' claim that "the meaning of the concept of model is the same in mathematics and the empirical sciences" (1969, 12). Several philosophers of science have pursued the idea of using an informal version of model-theoretic models for scientific modelling. Sometimes the models are described as non-linguistic - this might be hard to reconcile with our definition of models in section 1 above.

Cognitive science is one area where the difference between models and modelling tends to become blurred. A central question of cognitive science is how we represent facts or possibilities in our minds. If one formalises these mental representations, they become something like 'models of phenomena'. But it is a serious hypothesis that in fact our mental representations have a good deal in common with simple set-theoretic structures, so that they are 'models' in the model-theoretic sense too. In 1983 two influential works of cognitive science were published, both under the title Mental Models. The first, edited by Dedre Gentner and Albert Stevens, was about people's 'conceptualizations' of the elementary facts of physics; it belongs squarely in the world of 'modelling of phenomena'. The second, by Philip Johnson-Laird, is largely about reasoning, and makes several appeals to 'model-theoretic semantics' in our sense. Researchers in the Johnson-Laird tradition tend to refer to their approach as 'model theory', and to see it as allied in some sense to what we have called model theory.

Pictures and diagrams seem at first to hover in the middle ground between theories and models. In practice model theorists often draw themselves pictures of structures, and use the pictures to think about the structures. On the other hand pictures don't generally carry the labelling that is an essential feature of model-theoretic structures. There is a fast growing body of work on reasoning with diagrams, and the overwhelming tendency of this work is to see pictures and diagrams as a form of language rather than as a form of structure. For example Eric Hammer and Norman Danner (1996) describe a 'model theory of Venn diagrams'; the Venn diagrams themselves are the syntax, and the model theory is a set-theoretical explanation of their meaning. (A curious counterexample is the horizontal line diagrams of the 12th century Baghdad Jewish scholar Abū l-Barakāt they represent structures and not propositions, and Abū l-Barakāt uses them to express model-theoretic consequence in syllogisms. Further details are in Hodges 2018 on model-theoretic consequence.)

The model theorist Yuri Gurevich introduced abstract state machines (ASMs) as a way of using model-theoretic ideas for specification in computer science. According to the Abstract State Machine website (see Other Internet Resources below),

any algorithm can be modeled at its natural abstraction level by an appropriate ASM. ... ASMs use classical mathematical structures to describe states of a computation; structures are well-understood, precise models.

The book of Börger and Stärk cited below is an authoritative account of ASMs and their uses.

Today you can make your name and fortune by finding a good representation system. There is no reason to expect that every such system will fit neatly into the syntax/semantics framework of model theory, but it will be surprising if model-theoretic ideas don't continue to make a major contribution in this area.

The sections above considered some of the basic ideas that fed into the creation of model theory, noting some ways in which these ideas appeared either in mathematical model theory or in other disciplines that made use of model theory. None of this is particularly philosophical, except in the broad sense that philosophers work with ideas. But as mathematical model theory has become more familiar to philosophers, it has increasingly become a source of material for philosophical questions. In 2018 two books appeared that directly addressed this philosophical use of model theory, though in very different ways.

In the first book, Button and Walsh 2018, the authors present an invitation to the reader to help create a discipline, 'philosophy and model theory', which is gradually coming into existence. (This is partly belied by the large amount of carefully-worked material in the book.) Mathematics in general is a source of fundamental philosophical worries. For example mathematicians refer to entities that we have no causal interaction with (such as the number π or the set of real numbers), and this creates questions about how we can identify these entities to ourselves or each other, and how we can discover facts about them. These problems are not new or peculiar to model theory; but mathematical model theory is the part of mathematics most concerned with 'reference' and 'isomorphism types' and 'indiscernibility', notions which go directly to the philosophical problem areas. The authors give clear analyses of exactly what the issues are in key discussions in these areas.

The second book, Baldwin 2018, presents mathematical model theory of the period from 1970 to today as a source of material for the discipline of philosophy of mathematical practice. This discipline studies the work of particular mathematicians within their historical context, and asks such questions as: Why did this mathematician prefer classifications in terms of X to classifications in terms of Y? Why did this group of mathematical researchers choose to formalise their subject matter using such-and-such a language or set of symbols? How did they decide what to formalise and what to leave unformalised? The discipline is partly historical, but it looks for conceptual justifications of the historical choices made. (See the entries mathematical style and explanation in mathematics.) Baldwin has a long history of work in mathematical model theory, so he can answer questions like those above from personal knowledge. This book gives a rich supply of examples, explained with helpful pictures and remarkably little technical notation.

Introductory texts

Doets, K., 1996, Basic Model Theory, Stanford: CSLI Publications.

Hodges, W., 1997, A Shorter Model Theory, Cambridge: Cambridge University Press.

Manzano, M., 1999, Model Theory, Oxford: Oxford University Press.

Rothmaler, P., 2000, Introduction to Model Theory, Amsterdam: Gordon and Breach.

Model-theoretic definition

Frege, G., 1906, "Grundlagen der Geometrie", Jahresbericht der deutschen Mathematikervereinigung, 15: 293-309, 377-403, 423-430.

Gergonne, J., 1818, "Essai sur la théorie de la définition", Annales de Mathématiques Pures et Appliquées, 9: 1-35.

Hilbert, D., 1899, Grundlagen der Geometrie, Leipzig: Teubner.

Hodges, W., 2008, "Tarski's theory of definition", in Patterson, D. New Essays on Tarski and Philosophy, Oxford: Oxford University Press, pp. 94-132.

Lascar, D., 1998, "Perspective historique sur les rapports entre la théorie des modèles et l'algèbre", Revue d'histoire des mathématiques, 4: 237-260.

Mancosu, P., Zach, R. and Badesa, C., 2009, "The development of mathematical logic from Russell to Tarski", in L. Haaparanta (ed.), The Development of Modern Logic, Oxford: Oxford University Press, pp. 318-470.

Pasch, M., 1882, Vorlesungen über Neuere Geometrie, Berlin: Springer-Verlag.

Robinson, A., 1952, "On the application of symbolic logic to algebra", Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950, Volume 1), Providence, RI: American Mathematical Society, pp. 686-694.

Suppes, P., 1957, "Theory of definition" in Introduction to Logic (Chapter 8), Princeton, NJ: Van Nostrand.

Tarski, A., 1954, "Contributions to the theory of models, I", Indagationes Mathematicae, 16: 572-581.

Model-theoretic consequence

Blanchette, P., 1996, "Frege and Hilbert on consistency", The Journal of Philosophy, 93: 317-336.

---, 2012, Frege's Conception of Logic, New York: Oxford University Press.

Boole, G., 1847, The Mathematical Analysis of Logic, Cambridge: Macmillan, Barclay and Macmillan.

Etchemendy, J., 1990, The Concept of Logical Consequence, Cambridge, MA: Harvard University Press.

Frege, G., 1971, On the Foundations of Geometry, and Formal Theories of Arithmetic, E. Kluge (trans.), New Haven: Yale University Press.

Gómez-Torrente, M., 1996, "Tarski on logical consequence", Notre Dame Journal of Formal Logic, 37: 125-151.

Hodges, W. 2004, "The importance and neglect of conceptual analysis: Hilbert-Ackermann iii.3", in V. Hendricks et al. (eds.), First-Order Logic Revisited, Berlin: Logos, pp. 129-153.

---, 2018, "Two early Arabic applications of model-theoretic consequence", Logica Universalis, 12: 37-54.

Kreisel, G., 1969, "Informal rigour and completeness proofs", in J. Hintikka (ed.), The Philosophy of Mathematics, London: Oxford University Press, pp. 78-94.

Tarski, A., 1983, "On the concept of logical consequence", translated in A. Tarski, Logic, Semantics, Metamathematics, J. Corcoran (ed.), Indianapolis: Hackett, pp. 409-420.

van Benthem, J., 1991 [1983], The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse, Dordrecht: Reidel, 1983; second edition, Springer, 1991.

Expressive strength

Cutland, N., 2009, Nonstandard Analysis and its Applications, Cambridge: Cambridge University Press.

Ebbinghaus, H.-D., and Flum, J., 1999, Finite Model Theory, Berlin: Springer-Verlag.

Ebbinghaus, H.-D., Flum, J. and Thomas, W., 1984, Mathematical Logic, New York: Springer-Verlag.

Fleuriot, J., 2001, A Combination of Geometry Theorem Proving and Nonstandard Analysis, with Application to Newton's Principia, New York: Springer-Verlag.

Immerman, N., 1999, Descriptive Complexity, New York: Springer-Verlag.

Libkin, L., 2004, Elements of Finite Model Theory, Berlin: Springer-Verlag.

Lindström, P., 1969, "On extensions of elementary logic", Theoria, 35:1-11.

Loeb, P. and Wolff, M. (eds.), 2000, Nonstandard Analysis for the Working Mathematician, Dordrecht: Kluwer.

Robinson, A., 1967, "The metaphysics of the calculus", in Problems in the Philosophy of Mathematics, I. Lakatos (ed.), Amsterdam : North-Holland, pp. 28-40.

Models and modelling

Allwein, G. and Barwise, J. (eds.), 1996, Logical Reasoning with Diagrams, New York: Oxford University Press.

Börger, E. and Stärk, R., 2003, Abstract State Machines: A Method for High-Level System Design and Analysis, Berlin: Springer-Verlag.

Fowler, M., 2000, UML Distilled, Boston: Addison-Wesley.

Garnham, A., 2001, Mental Models and the Interpretation of Anaphora, Philadelphia: Taylor and Francis.

Gentner, D. and Stevens, A. (eds.), 1983, Mental Models, Hillsdale, NJ: Lawrence Erlbaum.

Gurevich, Yuri, 1993, "Evolving Algebras: An Attempt to Discover Semantics", in E. Börger (ed.), Specification and Validation Methods, pp. 9-36, Oxford: Oxford University Press.

Hammer, E. and Danner, N., 1996, "Towards a Model Theory of Venn Diagrams", in Allwein and Barwise (eds.) 1996, pp. 109-128.

Johnson-Laird, P., 1983, Mental Models: Towards a cognitive science of language, inference, and consciousness, Cambridge: Cambridge University Press.

Meijers, A. (ed.), 2009, Philosophy of Technology and Engineering Sciences, Amsterdam: Elsevier; see chapters W. Hodges, "Functional modelling and mathematical models"; R. Müller, "The notion of a model, theories of models and history"; and N. Nersessian, "Model based reasoning in interdisciplinary engineering".

Moktefi, A. and Shin, S.-J. (eds.), 2013, Visual Reasoning with Diagrams, Basel: Birkhäuser.

Morgan, M. S. and Morrison, M. (eds.), 1999, Models as Mediators, Cambridge: Cambridge University Press.

Pullum, G. K. and Scholz, B. C., 2001, "On the distinction between model-theoretic and generative-enumerative syntactic frameworks", in Logical Aspects of Computational Linguistics (Lecture Notes in Computer Science: Volume 2099), P. De Grooteet al. (eds.), Berlin: Springer-Verlag, pp. 17-43.

Stenning, K., 2002, Seeing Reason, Oxford: Oxford University Press.

Suppes, P., 1969, Studies in the Methodology and Foundations of Science, Dordrecht: Reidel.

Philosophy of model theory

Baldwin, J., 2018, Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism, Cambridge: Cambridge University Press.

Button, T. and Walsh, S., 2018, Philosophy and Model Theory, Oxford: Oxford University Press.

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Sometimes we write or speak a sentence \(S\) that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that \(S\) comes to express a true or false statement, we are said to \(S\), and the added information is called an interpretation of \(S\).

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https://www.youtube.com/watch?v=itrXYg41-V0

PhioGodinez_Explains the relation of two proposition and how to determine if the proposition is true or false

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INTRODUCTION to PROPOSITIONAL LOGIC - DISCRETE MATHEMATICS

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https://www.researchgate.net/publication/281838644_An_Introduction_to_Business_Mathematics

SarahBelleza_This link shows you a pdf file on the introduction of business mathematics. It can tell you about the basic concepts of each topic that is a part in business math.

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Miyoko Miniano
We can learn a lot on this website.

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Mathematical Logic (AND, OR & NOT) | Formulas and Examples

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The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

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Rhyme Godfrey Anque_you can download this file to know the basics of business mathematics

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https://www.vedantu.com/maths/business-mathematics
CheyneSalvatus_This link will help us understand the definition of Business Mathematics and its concepts.

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Learn about business mathematics topic of Maths in details explained by subject experts on vedantu.com. Register free for online tutoring session to clear your doubts.

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Bonifacio, Edclarence_
This video shows us the overview of what to expect in this topic and has been very helpful for me

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https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions

SarahBelleza_This link gives you a variety of topics about functions in mathematics. It also gives you videos to watch and tests to try regarding the topic.

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Functions | Algebra 1 | Math | Khan Academy

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In this unit, we learn about functions, which are mathematical entities that assign unique outputs to given inputs. We'll evaluate, graph, analyze, and create various types of functions.

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https://www.aiou.edu.pk/SoftBooks/1349.pdf
JaredJiongco_This PDF file shows topics and examples about business mathematics

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Rhyme Godfrey Anque_ basic introduction to logic of mathematica and other important information

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Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements such as -

  such that   where

Which in Simple English means "There exists an integer that is not the sum of two squares". Importance of Mathematical Logic The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments. Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of computer programs and verification of correctness of programs.

Propositional Logic

What is a proposition? A proposition is the basic building block of logic. It is defined as a declarative sentence that is either True or False, but not both. The Truth Value of a proposition is True(denoted as T) if it is a true statement, and False(denoted as F) if it is a false statement. For Example,

1. The sun rises in the East and sets in the West.
2. 1 + 1 = 2
3. 'b' is a vowel.

All of the above sentences are propositions, where the first two are Valid(True) and the third one is Invalid(False). Some sentences that do not have a truth value or may have more than one truth value are not propositions. For Example,

1. What time is it?
2. Go out and play.
3. x + 1 = 2.

The above sentences are not propositions as the first two do not have a truth value, and the third one may be true or false. To represent propositions, propositional variables are used. By Convention, these variables are represented by small alphabets such as . The area of logic which deals with propositions is called propositional calculus or propositional logic. It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators.

Truth Table

Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table. Most Common Logical Connectives-

1. Negation - If is a proposition, then the negation of is denoted by , which when translated to simple English means- "It is not the case that " or simply "not ". The truth value of is the opposite of the truth value of . The truth table of is-

Example, The negation of "It is raining today", is "It is not the case that is raining today" or simply "It is not raining today".

2. Conjunction - For any two propositions and , their conjunction is denoted by , which means " and ". The conjunction is True when both and are True, otherwise False. The truth table of is-

Example, The conjunction of the propositions - "Today is Friday" and - "It is raining today", is "Today is Friday and it is raining today". This proposition is true only on rainy Fridays and is false on any other rainy day or on Fridays when it does not rain.

3. Disjunction - For any two propositions and , their disjunction is denoted by , which means " or ". The disjunction is True when either or is True, otherwise False. The truth table of is-

Example, The disjunction of the propositions - "Today is Friday" and - "It is raining today", is "Today is Friday or it is raining today". This proposition is true on any day that is a Friday or a rainy day(including rainy Fridays) and is false on any day other than Friday when it also does not rain.

4. Exclusive Or - For any two propositions and , their exclusive or is denoted by , which means "either or but not both". The exclusive or is True when either or is True, and False when both are true or both are false. The truth table of is-

Example, The exclusive or of the propositions - "Today is Friday" and - "It is raining today", is "Either today is Friday or it is raining today, but not both". This proposition is true on any day that is a Friday or a rainy day(not including rainy Fridays) and is false on any day other than Friday when it does not rain or rainy Fridays.

5. Implication - For any two propositions and , the statement "if then " is called an implication and it is denoted by . In the implication , is called the hypothesis or antecedent or premise and is called the conclusion or consequence. The implication is is also called a conditional statement. The implication is false when is true and is false otherwise it is true. The truth table of is-

You might wonder that why is true when is false. This is because the implication guarantees that when and are true then the implication is true. But the implication does not guarantee anything when the premise is false. There is no way of knowing whether or not the implication is false since did not happen. This situation is similar to the "Innocent until proven Guilty" stance, which means that the implication is considered true until proven false. Since we cannot call the implication false when is false, our only alternative is to call it true. This follows from the Explosion Principle which says- "A False statement implies anything" Conditional statements play a very important role in mathematical reasoning, thus a variety of terminology is used to express , some of which are listed below.

"if , then  ""  is sufficient for  ""  when  ""a necessary condition for   is  ""  only if  ""  unless  ""  follows from  "

Example, "If it is Friday then it is raining today" is a proposition which is of the form . The above proposition is true if it is not Friday(premise is false) or if it is Friday and it is raining, and it is false when it is Friday but it is not raining.

6. Biconditional or Double Implication - For any two propositions and , the statement " if and only if(iff) " is called a biconditional and it is denoted by . The statement is also called a bi-implication. has the same truth value as The implication is true when and have same truth values, and is false otherwise. The truth table of is-

Some other common ways of expressing are-

"  is necessary and sufficient for  ""if   then  , and conversely""  iff  "

Example, "It is raining today if and only if it is Friday today." is a proposition which is of the form . The above proposition is true if it is not Friday and it is not raining or if it is Friday and it is raining, and it is false when it is not Friday or it is not raining. Exercise:

1) Consider the following statements:

  P: Good mobile phones are not cheap.
  Q: Cheap mobile phones are not good.
  L: P implies Q
  M: Q implies P
  N: P is equivalent to Q

Which one of the following about L, M, and N is CORRECT?(Gate 2014)

(A) Only L is TRUE.

(B) Only M is TRUE.

(C) Only N is TRUE.

(D) L, M and N are TRUE.

For solution, see GATE | GATE-CS-2014-(Set-3) | Question 11

2) Which one of the following is not equivalent to p⇔q (Gate 2015)

For solution, see GATE | GATE-CS-2015 (Set 1) | Question 65

References- Propositional Logic - Wikipedia Principle of Explosion - Wikipedia

Discrete Mathematics and its Applications, by Kenneth H Rosen

Read next part : Introduction to Propositional Logic - Set 2

This article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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Mathematics | Introduction to Propositional Logic | Set 1 - GeeksforGeeks

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Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements such as - such that where Which in Simple English means "There exists an integer that is not the sum of two squares".

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https://www.math.toronto.edu/preparing-for-calculus/3_logic/we_2_if_then.html
JaredJiongco_This website shows us some examples with statements and their solutions

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If...then... statements

In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B". The conditions that make up "A" are the assumptions we make, and the conditions that make up "B" are the conclusion.

If we are going to prove that the statement "If A, then B" is true, we would need to start by making the assumptions "A" and then doing some work to conclude that "B" must also hold.

If we want to apply a statement of the form "If A, then B", then we need to make sure that the conditions "A" are met, before we jump to the conclusion "B."

For example, if you want to apply the statement "$n$ is even $\Rightarrow$ $\frac{n}{2}$ is an integer", then you need to verify that $n$ is even, before you conclude that $\frac{n}{2}$ is an integer.

In mathematics you will often encounter statements of the form "A if and only if B" or "A $\Leftrightarrow$ B". These statements are really two "if/then" statements. The statement "A if and only if B" is equivalent to the statements "If A, then B" and "If B, then A." Another way to think of this sort of statement is as an equivalence between the statements A and B: whenever A holds, B holds, and whenever B hold, A holds.

Consider the following example: "$n$ is even $\Leftrightarrow \frac{n}{2}$ is an integer". Here the statement A is "$n$ is even" and the statement B is "$\frac{n}{2}$ is an integer." If we think about what it means to be even (namely that n is a multiple of 2), we see quite easily that these two statements are equivalent: If $n=2k$ is even, then $\frac{n}{2} = \frac{2k}{2} = k$ is an integer, and if $\frac{n}{2} = k$ is an integer, then $n=2k$ so $n$ is even.

In everyday use, a statement of the form "If A, then B", sometimes means "A if and only if B." For example, when most people say "If you lend me \$30, then I'll do your chores this week" they typically mean "I'll do your chores if and only if you lend me \$30." In particular, if you don't lend the \$30, they won't be doing your chores.

In mathematics, the statement "A implies B" is very different from "A if and only if B." Consider the following example: Let A be the statement "$n$ is an integer" and B be the statement "$\frac{n}{3}$ is a rational number." The statement "A implies B" is the statement "If $n$ is an integer, then $\frac{n}{3}$ is a rational number." This statement is true. However, the statement "A if and only if B" is the statement "$n$ is an integer if and only if $\frac{n}{3}$ is a rational number," which is false.

Mini-Lecture.

Below is a mini lecture about if-then statements.

Example.

Consider the statement "Suppose that it's raining. Then there is a cloud in the sky.".

(i)

Determine the hypotheses/assumptions and the conclusion.

(ii)

Rewrite this statement explicitly in the form "If A, then B" using Part (i).

(iii)

Is this statement true or false?

(i)

The hypothesis we are making is that it is raining. The conclusion we are making is that there must be a cloud in the sky.

(ii)

"If it's raining, then there must be a cloud in the sky."

(iii)

This statement is true. (Based on all that is currently known about how rain works!)

Example. Consider the statement "$x > 0 \Rightarrow x+1>0$". Is this statement true or false?

Solution. To determine it's truth value, first we look at the hypothesis: $x>0$. Whatever we want to conclude, it is a consequence of the fact that $x$ is positive.

Next, we look at the conclusion: $x+1>0$. This statement must be true, since $x+1 > x > 0$.

This means that the statement is true.

Example. Consider the statement "If $x$ is a positive integer or a solution to $x+3>4$, then $x>0$ and $x> \frac{1}{2}$." Is this statement true?

Solution. To determine if it's true, let's look first at the assumptions. We are assuming that either $x$ is a positive integer, or that it solves the inequality $x+3>4$.

Next let's consider the conclusion. We are concluding that $x$ must satisfy both inequalities $x>0$ and $x > \frac{1}{2}$. If we look more closely, we see that once we satisfy the second inequality, the first is redundant. (If $x>\frac{1}{2}$, then it must already be larger than zero.)

Now, in order for this statement to be true, we need that if $x$ solves either of the assumptions, then it must solve $x>\frac{1}{2}$. Well, the first assumption is that $x$ is a positive integer, which means that $x\geq 1$, so in this case the conclusion holds. The second assumption is that $x+3>4$, or equivalently, that $x>1$, which means the conclusion holds as well.

Example. Consider the statement "$0>1 \Rightarrow \sin x = 2$". Is this statement true or false?

Solution. To determine it's truth value, first we look at the hypothesis: $0>1$. This is obviously false!

So the statement is true! (Why?) This kind of statements "A $\Rightarrow$ B" where A is false are called vaccuously true.

A statement "A $\Rightarrow$ B" is true when the relation "A implies B" is true, not when A, or B, or A and B are true. It states that "if A is true, then B must also be true".

This means that when A is false, the statement doesn't conclude anything.

So whenever the hypothesis A is false, a statement "A $\Rightarrow$ B" is always true! (independently of whether B is true or false)

# data[].attributes.attachment_link.description
In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B".

# data[].attributes.body
https://www.vedantu.com/maths/business-mathematics

It helps/guides you to know the formulas, measurements, calculations, and etc. which can help us to complete other business tasks. This is also to provides us solution to business problems.

Domingo, Ma. Paulina Beatriz D.

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Learn about business mathematics topic of Maths in details explained by subject experts on vedantu.com. Register free for online tutoring session to clear your doubts.

# data[].attributes.body
Business math is a type of mathematics course that is meant to teach people about money and business mathematics are mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis.

https://en.wikipedia.org/wiki/Business_mathematics Christian Paul A. Torrico_ Key Takeaway

# data[].attributes.attachment_link.readable_text
Business mathematics are mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis.

Mathematics typically used in commerce includes elementary arithmetic, elementary algebra, statistics and probability. For some management problems, more advanced mathematics - calculus, matrix algebra, and linear programming - may be applied.

Business mathematics, sometimes called commercial math or consumer math, is a group of practical subjects used in commerce and everyday life. In schools, these subjects are often taught to students who are not planning a university education. In the United States, they are typically offered in high schools and in schools that grant associate's degrees; elsewhere they may be included under business studies. The emphasis in these courses is on computational skills and their practical application, with practice being predominant. These courses often fulfill the general math credit for high school students.

A (U.S.) business math course typically includes a review of elementary arithmetic, including fractions, decimals, and percentages. Elementary algebra is often included as well, in the context of solving practical business problems. The practical applications typically include checking accounts, price discounts, markups and Markup, payroll calculations, simple and compound interest, consumer and business credit, and mortgages and revenues.

Undergraduate

Business mathematics comprises mathematics credits taken at an undergraduate level by business students. The course is often organized around the various business sub-disciplines, including the above applications, and usually includes a separate module on interest calculations; the mathematics comprises mainly algebraic manipulation. Many programs, as mentioned, extend to more sophisticated mathematics. Common credits are Business Calculus andBusiness Statistics. Programs may also cover matrix operations, as above. At many US universities, business students (instead) study " finite mathematics", a course combining several applicable topics, including basic probability theory, an introduction to linear programming, some theory of matrices, and introductory game theory; the course sometimes includes a high level treatment of calculus.

Since these courses are focused on problems from the business world, the syllabus is then adjusted re standard courses in the mathematics or science fields. The calculus course especially emphasizes differentiation, leading to optimization of costs and revenue. Integration is less emphasized, as its business applications are fewer - used here in some interest calculations, and for (theoretically) aggregating costs and / or revenue - and it is more technically demanding. Relatedly, in a regular calculus course students study trigonometric functions, whereas courses here do not typically cover this area. As regards formal theory, as these are applied courses, very few proofs or derivations are included, unlike standard courses. (Although see Bachelor of Science in Business Administration and Bachelor of Business Science.) The statistics syllabus, similarly of an applied nature, is described under that article.

Note that economics majors, especially those planning to pursue graduate study in the field, are encouraged to instead take regular calculus, as well as linear algebra and other advanced math courses, especially real analysis. Some economics programs (instead) include a module in "mathematics for economists", providing a bridge between the above "Business Mathematics" courses and mathematical economics and econometrics. Programs in operations management, risk management, and management accounting, may similarly include supplementary coursework in relevant quantitative techniques, generally regression, and often linear programming as above, as well as other optimization techniques and various probability topics.

Postgraduate

At the postgraduate level, generalist management and finance programs include quantitative topics which are foundational for the study in question - often exempting students with an appropriate background. These are usually "interest mathematics" and statistics, both at the above level. MBA programs often also include basic operations research (linear programming, as above) with the emphasis on practice, and may combine the topics as "quantitative analysis"; MSF programs may similarly cover applied / financial econometrics.

More technical Master's in these areas, such as those in management science and quantitative finance, will entail a deeper, more theoretical study of operations research and econometrics, and extend to further advanced topics such as mathematical optimization and stochastic calculus. These programs do not include or entail "Business mathematics" per se.

Where mathematical economics is not a degree requirement, graduate economics programs often include "quantitative techniques", which covers (applied) linear algebra, multivariate calculus, and optimization, and may include dynamical systems and analysis; regardless, econometrics is usually a separate course, and is dealt with in depth.

Anthony , M. & Biggs, N. (1996). Mathematics for Economics and Finance: Methods And Modelling, Cambridge University Press. ISBN 9780521559133

Bradley, Teresa (2013). Essential Mathematics for Economics and Business 4th Edition, Wiley. ISBN 978-1118358290

Brechner, Robert. (2006). Contemporary Mathematics for Business and Consumers, Thomson South-Western. ISBN 0-324-30455-2

Dowling, Edward (2009). Schaum's Outline of Mathematical Methods for Business and Economics, McGraw-Hill. ISBN 0071635327

Rosser, M. & Lis, P. (2016). Basic Mathematics for Economists 3rd Edition. Routledge. ISBN 9780415485920

Wegner, Trevor. (2016). Applied Business Statistics: Methods and Excel-Based Applications, Juta Academic. ISBN 9781485111931

# data[].attributes.attachment_link.description
Business mathematics are mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis. Mathematics typically used in commerce includes elementary arithmetic, elementary algebra, statistics and probability. For some management problems, more advanced mathematics - calculus, matrix algebra, and linear programming - may be applied.

# data[].attributes.body
https://youtu.be/b5yjTKtr-Go
CheyneSalvatus_This video will help us understand logic and will teach us how to do it.

# data[].attributes.attachment_link.title
Introduction to Logic (Tagalog/Filipino Math)

# data[].attributes.body
https://byjus.com/maths/mathematical-logic/

This link provides a brief overview of mathematical logic.

# data[].attributes.attachment_link.title
Mathematical Logic (AND, OR & NOT) | Formulas and Examples

# data[].attributes.attachment_link.description
The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

# data[].attributes.headline
TYPES OF FUNCTIONS & THE DIFFERENT TYPES OF FUNCTIONS 

# data[].attributes.subject
TYPES OF FUNCTIONS & THE DIFFERENT TYPES OF FUNCTIONS 

# data[].attributes.body
https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.cuemath.com/algebra/types-of-functions/%23:~:text%3DThe%2520types%2520of%2520functions%2520can,and%2520onto%2520function%252C%2520into%2520function.&ved=2ahUKEwjBoryo6uD5AhXgpVYBHRToCqIQFnoECB0QBQ&usg=AOvVaw222oA_mXlleHemGShqegI3

John Carlo G Paguntalan
•The types of functions are defined on the basis of the domain, range, and function expression.Based on range, domain, equation, and the set of elements, there are the four categories of functions listed below.

# data[].attributes.body
Bonifacio, Edclarence_
In this module you can find and study about the basics of Business Math! It is very helpful especially when trying to figure out what this topic has in store for us.

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Citation preview

11 Business Mathematics Quarter 1 - Module 1: Expressing Fractions to Decimals and Percent and Vice Versa

Business Mathematics - Grade 11 Self-Learning Module (SLM) Quarter 1 - Module 1: Expressing Fractions to Decimals and Percent and Vice Versa First Edition, 2020

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Development Team of the Module Writer: Rhilen A. Tuardon Editors: Sally A. Palomo, Aillen S. Hiponia, Joecel S. Rubinos Reviewer: Zaida N. Abiera, Janice C. Antonio Illustrator: Maria Angelica T. Garcia Layout Artist: Sherwin P. Uy Cover Art Designer: Ian Caesar E. Frondoza Management Team: Allan G. Farnazo, CESO IV - Regional Director Fiel Y. Almendra, CESO V - Assistant Regional Director Romelito G. Flores, CESO V - Schools Division Superintendent Mario M. Bermudez, CESO VI - Assist. Schools Division Superintendent Gilbert B. Barrera - Chief, CLMD Arturo D. Tingson Jr. - REPS, LRMS Peter Van C. Ang-ug - REPS, ADM Jade T. Palomar - REPS, Mathematics Juliet F. Lastimosa - CID Chief Sally A. Palomo - Division EPS In- Charge of LRMS Gregorio O. Ruales - Division ADM Coordinator Zaida N. Abiera - Division EPS, Mathematics

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11 Business Mathematics Quarter 1 - Module 1: Expressing Fractions to Decimals and Percent and Vice Versa

Introductory Message For the facilitator: Welcome to the Business Mathematics for Grade 11 Self-Learning Module (SLM) on Expressing Fractions to Decimals and Percent and Vice Versa! This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module:

Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

2

For the learner: Welcome to the Business Mathematics - Grade 11 Self-Learning Module (SLM) on Expressing Fractions to Decimals and Percent and Vice Versa! The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This module has the following parts and corresponding icons: What I Need to Know

This will give you an idea of the skills or competencies you are expected to learn in the module.

What I Know

This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module.

What's In

This is a brief drill or review to help you link the current lesson with the previous one.

What's New

In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation.

What is It

This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills.

What's More

This comprises activities for independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module.

What I Have Learned

This includes questions or blank sentence/paragraph to be filled in to process what you learned from the lesson.

What I Can Do

This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns.

3

Assessment

This is a task which aims to evaluate your level of mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts.

Answer Key

This contains answers to all activities in the module.

At the end of this module you will also find:

References

This is a list of all sources used in developing this module.

The following are some reminders in using this module: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don't forget to answer What I Know before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!

4

What I Need to Know This module was designed and written with you in mind. It is here to help you master the expressing fractions to decimals and percent and vice versa. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. In this module, you will be able to:  express: a. fractions to decimal and percent forms; b. decimals to fractions and percent forms; and, c. percent to fractions and decimal forms.

5

What I Know Before we start this module, let us check first what you have learned about expressing fractions to decimals and percent and vice versa by answering the questions below. Direction: Encircle the letter of the correct answer. 1. What is the decimal form of a. b. c. d.

1 4

?

0.20 0.25 0.5 0.75

2. Which of the following is equivalent to a. b. c. d.

0.625 0.825 6.250 65.20

3. What is the percent form of a. b. c. d.

1 4

?

?

20% 25% 30% 40%

5. What is the percent form of a. b. c. d.

2

25% 30% 50% 75%

4. Which is the percent for a. b. c. d.

1

17 40

?

0.425% 4.25% 42.5% 425%

6

5 8

?

6. Which is the fraction for 0.7? a. b. c. d.

7 10 7 100 7 1000 70 1000

7. What is 0.55 as a fraction in lowest term? a. b. c. d.

55 100 11 20 55 1000 5 10

8. Which is the fraction for 0.125? a. b. c. d.

9. Which a. b. c. d.

3 8

1 8 3 5 1 7

of the following is equivalent to 0.65? 650% 6.5% 65% 0.65%

10. Express 4.5 in percent form. a. 0.45% b. 4.5% c. 45% d. 450%

7

11. Express 95% as a fraction in lowest term. a. b. c. d.

16 20 17 20 18 20 19 20

12. A shirt is discounted at 30% of its regular price. Which value is equivalent to 30%? a. b. c. d.

1 3 2 3 3 10 3 100

13. If you were to convert 33% in decimal form, what does it looks like? a. 0.33 b. 0.333 c. 0.033 d. 0.0033 14. Due to Covid-19 pandemic, only 85% of the previous students enrolled in a particular school. What is 85% in decimal form? a. 0.85 b. 0.085 c. 0.0085 d. 0.00085 15. In a certain section, 40% of the students preferred modular learning modalities. What is 40% in decimal form? a. 0.4 b. 0.04 c. 0.004 d. 0.0004 Great job! You have finally gauged your knowledge about expressing fractions to decimals and percent and vice versa.

8

Lesson

1

Expressing Fractions to Decimals and Percent and Vice Versa

In this module, you will learn about the concepts of fractions, decimals and percent. All numbers involving parts of a whole can be represented using fractions, decimals, and percent related to business.

What's In Hello! How are you? I am glad that you are indeed excited to take this module as you start this subject by reviewing what you have learned on expressing fractions, decimals and percent. It is necessary for us to learn how to express one of these forms to another. So, let us recall on the following terms for you to better understand the lessons in this module. A. CONCEPT OF FRACTION A fraction is a number written in the form

𝐚 𝐛

, where a and b are whole numbers

but b cannot be zero. The number on top, a is called numerator, and the number below, b, is called denominator.

In a fraction, the denominator tells the number of parts a whole is divided, while the numerator indicates how many of those are taken. The fraction

2 5

means the unit

is divided into five parts and that two of the five parts are taken. Moreover, means 2 ÷ 5.

9

2 5

also

Illustrative Examples: Kinds of Fractions

Examples

Proper fraction is a fraction less than one. Its numerator is less than its denominator.

1

Improper fraction is a fraction greater than or equal to one. Its numerator is greater than its denominator. Mixed number is the sum of a whole number and a proper fraction. Similar

fractions are fractions denominators

Dissimilar

with

fractions are fractions denominators.

the

with

, 2

5

, 6

10

5

7

12

5

,

3

,

9

4

1

1

3

5

2

4

2 , 4 , 6

same

different

1

3

9

9

,

1

, 4

,

7

, 5

5

3

9

8

B. CONCEPT OF DECIMAL A decimal is a representation of a fraction whose denominator is a multiple of 10.

Illustrative Example: Fraction

Decimal

1 10 2 100 456 100

0.1 0.02 56

= 4 100

4.56

104 1000

0.104

37 10000

0.0037

10

C. CONCEPT OF PERCENT A percent which means "per hundred" is a representation of a fraction whose denominator is 100. Thus, expressing a number in percent is like comparing it with 100.

Illustrative Example: a. 25% =

25 100

b. 88% =

88 100

c. 3% =

3 100

What's New Great Job! You have learned in your previous lesson about the concepts of fractions, decimals and percent. This time, you will learn more about this as we continue to express fractions to decimals and percent and vice versa. Before we proceed with the discussions, do the following activity to check your understanding on the basic concepts of fractions, decimals and percent. Activity 1: Complete Me! Direction: Complete the table below. Write the corresponding equivalents in the box. Fraction 1. 2. 3.

Decimal

Percent

𝟏 𝟓 𝟑 𝟐𝟎 𝟗 𝟏𝟎𝟎

4.

0.4

5.

0.12

6.

5.4

7.

34%

8.

115%

9.

4.5% Alright! Keep the fire burning! 11

What is It Fantastic! The previous activity gave us the idea that we can express fractions, decimals and percent in other forms. So now, let us discuss how to express fractions to decimals and percent and vice versa. A. EXPRESSING FRACTIONS TO DECIMALS AND PERCENT FORMS I. How to express fractions to decimals and percent forms? a. Express fraction to a decimal. b. Express decimal to a percent. Illustrative Examples: Problem 1: Express

4 5

as a decimal and as a percent.

Solution: 4

Write as a decimal.

5

Write as a percent.

=

Divide the numerator denominator. 4 ÷ 5 = 0.8

4÷5

𝟒 𝟓

the

To write the decimal as a percent, move the decimal point two places to the right.

0.8 = 80%

Answer

by

= 0.8 = 80%

Problem 2: Express 3 Solution:

7 as a decimal and as a percent. 8

7

38 = 3 + Write as a decimal.

7 8

=

7 8

7÷8

3 + 0.875 = 3.875 Write as a percent.

Answer

3.875 = 387.5% 𝟕

𝟑𝟖

Write the mixed fraction as 3 wholes plus the fractional part. Write the fractional part as a decimal by dividing the numerator by the denominator. 7 ÷ 8 = 0.875 Add 3 to the decimal. Now, you can move the decimal as a percent, move the decimal point two places to the right and write the percent (%) sign.

= 3.875 = 38.5% 12

B. EXPRESSING DECIMALS TO FRACTIONS AND PERCENT FORMS I. How to express decimals to fractions and percent forms? a. Express decimal to a percent. b. Express percent to a fraction. Illustrative Examples: Problem 1: Express 0.4 as a percent and as a simplified fraction. Solution:

Write as a percent

Write 0.4 as 0.40, which is 40 hundredths. 40 hundredths is 40 percent.

0.4 = 0.40 = 40%

0.4 =

You can also move the decimal point two places to the right to find the percent equivalent. To write 0.4 as a fraction, you read the decimal, 4 tenths, and write 4 tenths in fraction form.

4 10

Write as a fraction 4 10

Answer

=

4÷2 10 ÷2

=

0.4 = 40% =

Simplify the fraction by dividing the numerator and denominator by 2, a greatest common factor (GCF).

2 5 𝟐 𝟓

Problem 2: Express 3.14 as a percent and as a simplified fraction. Solution: Write as a percent.

Write as a fraction.

3.14 = 314%

314% = 314 100

Answer

Move the decimal point two places to the right to find the percent equivalent. Replace the percent sign by a denominator of 100.

=

314

100 314÷2

100 ÷2

=

157 50

3.14 = 314% = 𝟑

13

=3

𝟕 𝟓𝟎

7 50

Reduce the fraction to lowest term.

C. EXPRESSING PERCENT TO FRACTIONS AND DECIMAL FORMS I. How to express percent to fractions and decimal forms? a. Express percent to a fraction. b. Express fraction to a decimal. Illustrative Example Problem 1: Express 25% as a simplified fraction and as a decimal. Solution:

25% =

Since % means "out of 100," 25% means 25 out of 100. You write this as a fraction, using 100 as the denominator.

25 100

Write as a fraction 25 100

Write as a decimal

Answer

=

25% =

25 ÷25 100 ÷25 25 100

25% =

𝟏 𝟒

=

=

1

=

4

0.25

Simplify the fraction by dividing the numerator and denominator by the common factor 25. Move the decimal point in the whole number 25 two places to the left to get 0.25.

0.25

Problem 2: Express 195% as a simplified fraction and as a decimal. Solution: 195% = Write as a fraction

195 100

Write as a decimal

Answer

=

195 ÷5 100 ÷5

195% =

Replace the percent sign by a denominator of 100.

195 100

=

195 100

195% = 𝟏

𝟏𝟗 𝟐𝟎

39 20

or 1

=

1.95

=

1.95

14

19 20

Reduce the fraction to lowest term. Change the improper fraction to a mixed number. Move the decimal point in the whole number 195 two places to the left to get 1.95.

What's More Alright! We have already discussed the steps on how to express fractions to decimals and percent and vice versa. At this moment, be ready for our next activity. Here, you will be able to use what you have learned from our previous discussions. You can do this! Activity 2: Fill Me! Direction: Fill in the empty boxes with the correct percent, decimal, or fraction. Always simplify your answer. PERCENT

1.

0.81

144%

𝟗 𝟏𝟎

4.

5.

FRACTION

32%

2.

3.

DECIMAL

1.25

𝟐𝟒 𝟐𝟓

6.

15

What I Have Learned Awesome! Now, let us check your understanding based on the discussions about expressing fractions to decimals and percent and vice versa. Answer all the activities below and let us see how much you have learned. Activity 3: Fill Me! Direction: Fill each blank with the correct word to make the statement correct. Write your answer on the blank provided. 1. A number representing a part of a _________________ is called a fraction. 2. A percent is a ratio of some number to ____________. 3. To change a fraction to a percent, first divide the numerator by the denominator to obtain a decimal number. Then multiply the decimal number by 100 and add a ______________sign. 4. To change a decimal number to a percent, multiply the decimal number by ________________ and add a percent sign. 5. To change a percent to a decimal number, divide the number by 100 and ______________ the percent sign.

16

What I Can Do Well done! Now, here is another activity that will enable you to apply the lesson on expressing fractions to decimals and percent and vice versa by solving real-life problems. It is time to test and apply the knowledge from the discussions and activities. Activity 4: Answer Me! Direction: Read and solve each problem carefully. Write your answer in the box adjacent to the given problem. Given Problem

1. Richard spends

1 4

Solutions

of his monthly salary

on house rent and

3 5

of his monthly

salary on food, utilities, and transportation. He saves the rest. What part of his monthly salary does he save?

2. Joey has PhP286,300.00 in his bank account. If he buys a laptop worth PhP21,499.95, a bag worth PhP3,499.75, and an external hard drive worth PhP1,699.50, how much will be left in his account?

3. Sarah got 35 correct answers on their 50 - item Math Test. What percent of the test did she answer correctly?

17

Assessment Bravo! You have come this far. Before we end this module, let us evaluate what you have learned about expressing fractions to decimals and percent and vice versa by answering the following questions below. Direction: Read and understand each item very carefully. Then, encircle the letter of the correct answer. 1. What is the decimal form of a. b. c. d.

3 5

?

0.6 0.65 0.7 0.75

2. Which of the following is equivalent to a. b. c. d.

0.32 0.34 0.36 0.40 4

3. What is the percent form of a. b. c. d.

3 8

?

0.375% 3.75% 37.5% 375%

5. What is the percent form of a. b. c. d.

?

4% 40% 400% 4000%

4. Which is the percent for a. b. c. d.

10

21

?

30

58% 60% 65% 70%

18

9 25

?

6. Which is the fraction for 0.32? a. b. c. d.

1 8

3 20 8 25 2 3

7. What is 0.65 as a fraction in lowest term? a. b. c. d.

13 20 13 40 65 100 65 1000

8. Which is the fraction for 0.434? a. b. c. d. 9. Which a. b. c. d.

216 500 217 500 218 500 219 500

of the following is equivalent to 0.35? 3.5% 35% 350% 355%

10. Express 8.2 in percent form. a. 820% b. 82% c. 8.2% d. 0.82%

19

11. Express 35% as a fraction in lowest term.

a. b. c. d.

7 20 8 20 9 20 12 20

12. A camera is discounted at 60% of its regular price. Which value is equivalent to 60%?

a.

3

b.

3

c.

3

d.

3

4 5 6 7

13. If you were to convert 99% in decimal form, what does it looks like? a. 0.9 b. 0.99 c. 9.9 d. 9.99 14. Due to Covid-19 pandemic, only 24% of the previous students did not enrolled in a particular school. What is 24% in decimal form? a. 0.24 b. 0.024 c. 0.0024 d. 0.00024 15. If you a. b. c. d.

were to convert 70% in decimal form, what does it looks like? 0.7 0.07 0.007 0.0007

Great work in finishing the quiz! You have worked really hard to get this far. You are amazing!

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Additional Activities Congratulations, you made it! I know you have learned a lot from this module about expressing fractions to decimals and percent and vice versa. But for more additional activities, let us apply these skills in real-life situations. Direction: Read and analyze the problem carefully. Answer it by showing your complete solutions. Problem: Mary treated her friends Trisha, Ellen, and Beatrice in a pizza parlor. She ordered 2 large pizzas. Each pizza was sliced into 12. If the girls ate

3 4

of the pizzas and Mary brought home the

remaining slices for her sisters, how many slices of pizza did her sisters receive? Express your answer in fraction, decimal and percent form. Solutions:

Excellently Done! You did well on this module! Keep going.

Here are your 3 stars for a job well done. You are now ready to answer the next module on solving problems involving fractions, decimals, and percent.

21

22 Business Mathematics Teaching Guide pp. 36 - 37; 44 - 45 Lopez, B.R., Lundag, L. Dagal, K.A. & Garces, I.J. (2016). Business Math Textbook. Quezon City: Vibal Group, Inc. pp. 2 - 44

References What 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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Answer Key

DISCLAIMER This Self-Learning Module (SLM) was developed by DepEd SOCCSKSARGEN with the primary objective of preparing for and addressing the new normal. Contents of this module were based on DepEd's Most Essential Learning Competencies (MELC). This is a supplementary material to be used by all learners of Region XII in all public schools beginning SY 2020-2021. The process of LR development was observed in the production of this module. This is version 1.0. We highly encourage feedback, comments, and recommendations. For inquiries or feedback, please write or call: Department of Education - SOCCSKSARGEN Learning Resource Management System (LRMS) Regional Center, Brgy. Carpenter Hill, City of Koronadal Telefax No.: (083) 2288825/ (083) 2281893 Email Address: [email protected]

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Citation preview 11 Business Mathematics Quarter 1 - Module 1: Expressing Fractions to Decimals and Percent and Vice Versa Business Mathematics - Grade 11 Self-Learning Module (SLM) Quarter 1 - Module 1: Expressing Fractions to Decimals and Percent and Vice Versa First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines.

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https://www.cuemath.com/algebra/types-of-functions/

This link can give you insights about different types of functions as well as their definitions.

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The types of functions are defined on the basis of the domain, range, and function expression. The expression used to write the function is the prime defining factor for a function. Along with expression, the relationship between the elements of the domain set and the range set also accounts for the type of function. The classification of functions helps to easily understand and learn the different types of functions.

Every mathematical expression which has an input value and a resulting answer can be conveniently presented as a function. Here we shall learn about the types of functions and their definition, examples.

What are the Different Types of Functions?

The function y = f(x) is classified into different types of functions, based on factors such as the domain and range of a function, and the function expression. The functions have a domain x value that is referred as input. The domain value can be a number, angle, decimal, fraction. Similarly, the y value or the f(x) value (is generally a numeric value) is the range. The types of functions have been classified into the following four types.

Based on the Set Elements

Based on Equation

Based on Range

Based on Domain

Representation of Functions

There are three different forms of representation of functions. The functions need to be represented to showcase the domain values and the range values and the relationship between them. The functions can be represented with the help of Venn diagrams, graphical formats, and roster forms. The details of each of the forms of representation are as follows.

Venn Diagram: The Venn diagram is an important format for representing the function. The Venn diagrams are usually presented as two circles with arrows connecting the element in each of the circles. The domain is presented in one circle and the range values are presented in another circle. And the function defines the arrows, and how the arrows connect the different elements in the two circles.

Graphical Form: Functions are easy to understand if they are represented in the graphical form with the help of the coordinate axes. Representing the function in graphical form, helps us to understand the changing behavior of the functions if the function is increasing or decreasing. The domain of the function - the x value is represented along the x-axis, and the range or the f(x) value of the function is plotted with respect to the y-axis.

Roster Form: Roster notation of a set is a simple mathematical representation of the set in mathematical form. The domain and range of the function are represented in flower brackets with the first element of a pair representing the domain and the second element representing the range. Let us try to understand this with the help of a simple example. For a function of the form f(x) = x ², the function is represented as {(1, 1), (2, 4), (3, 9), (4, 16)}. Here the first element is the domain or the x value and the second element is the range or the f(x) value of the function.

List of Types of Functions

The types of functions are classified further to help for easy understanding and learning. The types of functions have been further classified into four different types, and are presented as follows.

Based on Elements

1. One One Function

2. Many One Function

3. Onto Function

4. One One and Onto Function

5. Into Function

6. Constant Function

Based on Equation

1. Identity Function

2. Linear Function

3. Quadratic Function

4. Cubic Function

5. Polynomial Functions

Based on the Range

1. Modulus Function

2. Rational Function

3. Signum Function

4. Even and Odd Functions

5. Periodic Functions

6. Greatest Integer Function

7. Inverse Function

8. Composite Functions

Based on the Domain

1. Algebraic Functions

2. Trigonometric Functions

3. Logarithmic Functions

Types of Functions - Based on Set Elements

These types of functions are classified based on the number of relationships between the elements in the domain and the codomain. The different types of functions based on set elements are as follows.

One One Function

A one-to-one function is defined by f: A → B such that every element of set A is connected to a distinct element in set B. The one-to-one function is also called an injective function. Here every element of the domain has a distinct image or co-domain element for the given function.

Many to One Function

A many to one function is defined by the function f: A → B, such that more than one element of the set A are connected to the same element in the set B. In a many to one function, more than one element has the same co-domain or image. If a many to one function, in the codomain, is a single value or the domain element are all connected to a single element, then it is called a constant function.

Onto Function

In an, onto function, every codomain element is related to the domain element. For a function defined by f: A → B, such that every element in set B has a pre-image in set A. The onto function is also called a subjective function.

One One and Onto Function (Bijection)

A function that is both a one and onto function is called a bijective function. Here every element of the domain is connected to a distinct element in the codomain and every element of the codomain has a pre-image. Also in other words every element of set A is connected to a distinct element in set B, and there is not a single element in set B which has been left out.

Into Function

Into function is exactly opposite in properties to an onto function. Here there are certain elements in the co-domain that do not have any pre-image. The elements in set B are excess and are not connected to any elements in set A.

Constant Function

A constant function is an important form of a many to one function. In a constant function, all the domain elements have a single image. The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain(x value), the same range value of K is obtained for a constant function.

Types of Function - Based on Equation

The algebraic expressions are also functions and are based on the degree of the polynomial. The functions based on equations are classified into the following equations based on the degree of the variable 'x'.

1. The polynomial function of degree zero is called a Constant Function.

2. The polynomial function of degree one is called a Linear Function.

3. The polynomial function of degree two is called a Quadratic Function.

4. The polynomial function of degree three is a Cubic Function.

Let us understand each of these functions in detail.

Identity Function

The identity function has the same domain and range. The identity function equation is f(x) = x, or y = x. The domain and range of the identity function is of the form {(1, 1), (2, 2), (3, 3), (4, 4).....(n, n)}.

The graph of the identity function is a straight line that is equally inclined to the coordinate axes and is passing through the origin. The identity function can take both positive and negative values and hence it is present in the first and the third quadrants of the coordinate axis.

Linear Function

A polynomial function having the first-degree equation is a linear function. The domain and range of a linear function is a real number, and it has a straight line graph. Equations such as y = x + 2, y = 3x, y = 2x - 1, are all examples of linear functions. The identity function of y = x can also be considered a linear function.

The general form of a linear function is f(x) = ax + b, is used to represent objective functions in linear programming problems. Here x, y are variables, and a, b are real numbers. Graphically the linear function can be represented by the equation of a line y = mx + c, where m is the slope of the line and c is the y-intercept of the line.

Quadratic Function

A quadratic function has a second-degree quadratic equation and it has a graph in the form of a curve. The general form of the quadratic function is f(x) = ax ² + bx + c, where a ≠ 0 and a, b, c are constant and x is a variable. The domain and range of the quadratic function is R.

The graph of a quadratic equation is a non-linear graph and is parabolic in shape. Examples of quadratic functions are f(x) = 3x ² + 5, f(x) = x ² - 3x + 2.

Cubic Function

A cubic function has an equation of degree three. The general form of a cubic function is f(x) = ax ³ + bx ² + cx +d, where a ≠ 0 and a, b, c, and d are real numbers & x is a variable. The domain and range of a cubic function is R.

The graph of a cubic function is more curved than the quadratic function. An example of cubic function is f(x) = 8x ³ + 5x ² + 3.

Polynomial Function

The general form of a polynomial function is f(x) = a nx n + a n-1x n-1 + a n-2x n-2+ ..... ax + b. Here n is a nonnegative integer and x is a variable. The domain and range of a polynomial function are R. Based on the power of the polynomial function, the functions can be classified as a quadratic function, cubic function, etc.

Types of Functions - Based on Range

Here the types of functions have been classified based on the range which is obtained from the given functions. The different types of functions based on the range are as follows.

Modulus Function

The modulus function gives the absolute value of the function, irrespective of the sign of the input domain value. The modulus function is represented as f(x) = |x|. The input value of 'x' can be a positive or a negative expression. The graph of a modulus function lies in the first and the second quadrants since the coordinates of the points on the graph are of the form (x, y), (-x, y).

Rational Function

A function that is composed of two functions and expressed in the form of a fraction is a rational function. A rational fraction is of the form f(x)/g(x), and g(x) ≠ 0. The functions used in this rational function can be an algebraic function or any other function. The graphical representation of these rational functions is similar to the asymptotes, since it does not touch the axis lines.

Signum Function

The signum function helps us to know the sign of the function and does not give the numeric value or any other values for the range. The range of the signum function is limited to {-1, 0, 1}. For the positive value of the domain, the signum function gives an answer of 1, for negative values the signum function gives an answer of -1, and for the 0 value of a domain, the image is 0. The signum function has wide application in software programming.

Even and Odd Function

The even and odd functions are based on the relationship between the input and the output values of the function. For the negative domain value, if the range is a negative value of the range of the original function, then the function is an odd function. And for the negative domain value, if the range is the same as that of the original function, then the function is an even function.

If f(-x) = f(x), for all values of x, then the function is an even function, and if f(-x) = -f(x), for all values of x, then the function is an odd function. An example of even functions are x ², Cosx, Secx, and an example of odd functions are x ³, Sinx, Tanx.

Periodic Function

The function is considered a periodic function if the same range appears for different domain values and in a sequential manner. The trigonometric functions can be considered periodic functions. For example, the function f(x) = Sinx, have a range [-1, 1] for the different domain values of x = nπ + (-1) n x. Similarly, we can write the domain and the range of the trigonometric functions and prove that the range shows up in a periodic manner.

Inverse function

The inverse of a function f(x) is denoted by f⁻¹(x). For inverse of a function the domain and range of the given function is changed as the range and domain of the inverse function. The inverse of a function can be prominently seen in algebraic functions and in inverse trigonometric functions. The domain of Sinx is R and its range is [-1, 1], and for Sin⁻¹ x the domain is [-1, 1] and the range is R. The inverse of a function exists, if it is a bijective function.

If a function f(x) = x ², then the inverse of the function is f⁻¹(x) = \(\sqrt x\).

Greatest Integer Function

The greatest integer function is also known as the step function. The greatest integer function rounds up the number to the nearest integer less than or equal to the given number. Clearly, the input variable x can take on any real value. However, the output will always be an integer. Also, all integers will occur in the output set. Thus, the domain of this function is real numbers R, while its range is integers (Z).

The greatest integer function graph is known as the step curve because of the step structure of the curve.The greatest integral function is denoted as f(x) = ⌊x⌋. For a function taking values from [1, 2), the value of f(x) is 1.

Composite Function

The composite functions are of the form of gof(x), fog(x), h(g(f(x))), and is made from the individual functions of f(x), g(x), h(x). The composite functions made of two functions have the range of one function forming the domain for another function. Let us consider a composite function fog(x), which is made up of two functions f(x) and g(x).

Here we write fog(x) = f(g(x)). The range of g(x) forms the domain for the function f(x). It can be considered as a sequence of two functions. If f(x) = 2x + 3 and g(x) = x + 1 we have fog(x) = f(g(x)) = f(x + 1) = 2(x + 1) + 3 = 2x + 5.

Types oF Functions - Based on Domain

Functions are used in all the other topics of maths. The functions have been classified based on the types of equations used to define the functions. The function equations generally have algebraic expressions, trigonometric functions, logarithms, exponents, and hence are named based on these domain values. The three broad types of functions based on domain value are as follows.

Algebraic Function

An algebraic function is helpful to define the various operations of algebra. The algebraic function has a variable, coefficient, constant term, and various arithmetic operators such as addition, subtraction, multiplication, division. An algebraic function is generally of the form of f(x) = a nx n + a n - 1x n - 1+ a n-2x n-2+ ....... ax + c.

The algebraic function can also be represented graphically. The algebraic function is also termed as a linear function, quadratic function, cubic function, polynomial function, based on the degree of the algebraic equation.

Trigonometric Functions

The trigonometric functions also have a domain and range similar to any other function. The six trigonometric functions are f(θ) = sinθ, f(θ) = cosθ, f(θ) = tanθ, f(θ) = secθ, f(θ) = cosecθ. Here the domain value θ is the angle and is in degrees or in radians. These trigonometric functions have been taken based on the ratio of the sides of a right-angle triangle, and are based on the Pythagoras theorem.

Further from these trigonometric functions, inverse trigonometric functions have also been derived. The domain of the inverse trigonometric function is a real number value and its range is an angle. The trigonometric functions and the inverse trigonometric functions are also sometimes referred to as periodic functions since the principal values are repeated.

Logarithmic Functions

Logarithmic functions have been derived from the exponential functions. The logarithmic functions are considered as the inverse of exponential functions. Logarithmic functions have a 'log' in the function and it has a base. The logarithmic function is of the form y = \(\log_ax \). Here the domain value is the input value of 'x' and is calculated using the Napier logarithmic table. The logarithmic function gives the number of exponential times to which the base has raised to obtain the value of x. The same logarithmic function can be expressed as an exponential function as x = a y.
Related Topic
The following topic help in a better understanding of the types of functions.

Solved Examples on Types of Functions
Book a Free Trial Class
FAQs on Types of Functions

What are the Types of Functions in Maths?

The types of functions can be broadly classified into four types.

Based on Element: One to one Function, many to one function, onto function, one to one and onto function, into function.

Based on Domain: Algebraic Functions, Trigonometry functions, logarithmic functions.

Based on Range: Modulus function, rational function, signum function, even and odd function, greatest integer function.

Based on Equations: Identity function, linear function, quadratic function, cubic function, polynomial function.

How do you Determine the Types of Functions?

The types of functions can be determined based on the domain, range, and functional equation. Further classifying functions into types of functions helps to group and easily understand each of the types of functions. All the trigonometric functions can be grouped under periodic functions. Functions such as identity function, linear function, the quadratic, cubic function can be grouped under polynomial functions. Thus the function equation y = f(x) is helpful to define the type of function.

What Type of Functions is Always Continuous?

The functions which have a range for every input value of the domain can be termed as a continuous function. The simplest example of a continuous function is the identity function which has equal domain and range value. Other examples of continuous functions are the trigonometric sine function and cosine functions.

What are the Various Forms of Representation of Functions?

The functions are generally represented in the form of an equation y = f(x), where x is the domain and y or f(x) is the range of the function. The functions with the domain and range elements are also represented as venn diagrams or as roster form. In roster form the domain and range of the function are represented as {\((x_1, f(x_1)), (x_2, f(x_2)), (x_3, f(x_3))\)}.

What are the other math applications of Types of Functions?

The types of functions have enormous applications in algebra, trigonometry, logarithms, exponents. All the algebraic expressions can be counted as functions as it has an input domain value of x and the output range, which is the answer of the algebraic function. The trigonometric functions also have the angle value as the domain and range value.

Can any Math Equation be Considered a Function?

Yes. A math equation that is not equal to zero can be considered as a function. Alternatively, a math equation with two variables where one variable can be taken as a domain and the other variable can be taken as the range, can be called a function.

What are the applications of types of functions?

The types of functions have numerous applications in physics, engineering, computer sciences, artificial intelligence. All of these fields aim at connecting one set of data points (domain) to another set of data points(range). Also, the functions help in representing the huge set of data points in a simple mathematical expression of the formal y = f(x).

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Types of Functions - Types, Definition, Examples

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The types of functions are defined on the basis of the domain, range, and function expression. The expression used to write the function is the prime defining factor for a function. Along with expression, the relationship between the elements of the domain set and the range set also accounts for the type of function.

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https://www.youtube.com/watch?v=q2eyZZK-OIk

dela Rosa, Rob Aileron R.

This video introduces us to logical statements by giving out the basic foundations of the topic.

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Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements such as -

  such that   where

Which in Simple English means "There exists an integer that is not the sum of two squares". Importance of Mathematical Logic The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments. Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of computer programs and verification of correctness of programs.

Propositional Logic

What is a proposition? A proposition is the basic building block of logic. It is defined as a declarative sentence that is either True or False, but not both. The Truth Value of a proposition is True(denoted as T) if it is a true statement, and False(denoted as F) if it is a false statement. For Example,

1. The sun rises in the East and sets in the West.
2. 1 + 1 = 2
3. 'b' is a vowel.

All of the above sentences are propositions, where the first two are Valid(True) and the third one is Invalid(False). Some sentences that do not have a truth value or may have more than one truth value are not propositions. For Example,

1. What time is it?
2. Go out and play.
3. x + 1 = 2.

The above sentences are not propositions as the first two do not have a truth value, and the third one may be true or false. To represent propositions, propositional variables are used. By Convention, these variables are represented by small alphabets such as . The area of logic which deals with propositions is called propositional calculus or propositional logic. It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators.

Truth Table

Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table. Most Common Logical Connectives-

1. Negation - If is a proposition, then the negation of is denoted by , which when translated to simple English means- "It is not the case that " or simply "not ". The truth value of is the opposite of the truth value of . The truth table of is-

Example, The negation of "It is raining today", is "It is not the case that is raining today" or simply "It is not raining today".

2. Conjunction - For any two propositions and , their conjunction is denoted by , which means " and ". The conjunction is True when both and are True, otherwise False. The truth table of is-

Example, The conjunction of the propositions - "Today is Friday" and - "It is raining today", is "Today is Friday and it is raining today". This proposition is true only on rainy Fridays and is false on any other rainy day or on Fridays when it does not rain.

3. Disjunction - For any two propositions and , their disjunction is denoted by , which means " or ". The disjunction is True when either or is True, otherwise False. The truth table of is-

Example, The disjunction of the propositions - "Today is Friday" and - "It is raining today", is "Today is Friday or it is raining today". This proposition is true on any day that is a Friday or a rainy day(including rainy Fridays) and is false on any day other than Friday when it also does not rain.

4. Exclusive Or - For any two propositions and , their exclusive or is denoted by , which means "either or but not both". The exclusive or is True when either or is True, and False when both are true or both are false. The truth table of is-

Example, The exclusive or of the propositions - "Today is Friday" and - "It is raining today", is "Either today is Friday or it is raining today, but not both". This proposition is true on any day that is a Friday or a rainy day(not including rainy Fridays) and is false on any day other than Friday when it does not rain or rainy Fridays.

5. Implication - For any two propositions and , the statement "if then " is called an implication and it is denoted by . In the implication , is called the hypothesis or antecedent or premise and is called the conclusion or consequence. The implication is is also called a conditional statement. The implication is false when is true and is false otherwise it is true. The truth table of is-

You might wonder that why is true when is false. This is because the implication guarantees that when and are true then the implication is true. But the implication does not guarantee anything when the premise is false. There is no way of knowing whether or not the implication is false since did not happen. This situation is similar to the "Innocent until proven Guilty" stance, which means that the implication is considered true until proven false. Since we cannot call the implication false when is false, our only alternative is to call it true. This follows from the Explosion Principle which says- "A False statement implies anything" Conditional statements play a very important role in mathematical reasoning, thus a variety of terminology is used to express , some of which are listed below.

"if , then  ""  is sufficient for  ""  when  ""a necessary condition for   is  ""  only if  ""  unless  ""  follows from  "

Example, "If it is Friday then it is raining today" is a proposition which is of the form . The above proposition is true if it is not Friday(premise is false) or if it is Friday and it is raining, and it is false when it is Friday but it is not raining.

6. Biconditional or Double Implication - For any two propositions and , the statement " if and only if(iff) " is called a biconditional and it is denoted by . The statement is also called a bi-implication. has the same truth value as The implication is true when and have same truth values, and is false otherwise. The truth table of is-

Some other common ways of expressing are-

"  is necessary and sufficient for  ""if   then  , and conversely""  if  "

Example, "It is raining today if and only if it is Friday today." is a proposition which is of the form . The above proposition is true if it is not Friday and it is not raining or if it is Friday and it is raining, and it is false when it is not Friday or it is not raining. Exercise:

1) Consider the following statements:

  P: Good mobile phones are not cheap.
  Q: Cheap mobile phones are not good.
  L: P implies Q
  M: Q implies P
  N: P is equivalent to Q

Which one of the following about L, M, and N is CORRECT?(Gate 2014)

(A) Only L is TRUE.

(B) Only M is TRUE.

(C) Only N is TRUE.

(D) L, M and N are TRUE.

For solution, see GATE | GATE-CS-2014-(Set-3) | Question 11

2) Which one of the following is not equivalent to p⇔q (Gate 2015)

For solution, see GATE | GATE-CS-2015 (Set 1) | Question 65

References- Propositional Logic - Wikipedia Principle of Explosion - Wikipedia

Discrete Mathematics and its Applications, by Kenneth H Rosen

Read next part : Introduction to Propositional Logic - Set 2

This article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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Mathematics | Introduction to Propositional Logic | Set 1 - GeeksforGeeks

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Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements such as - such that where Which in Simple English means "There exists an integer that is not the sum of two squares".

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https://byjus.com/maths/mathematical-logic/

The website provides basic information about different topics that relate to the topic of Logic. Such as set theory, model theory, proof theory, recursion theory, and some other definitions. Links that are related to this lesson can also be found on the website.

Pantas, Aicelle P.

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Mathematical Logic (AND, OR & NOT) | Formulas and Examples

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The basic mathematical logics are conjunction, disjunction, and negation are explained with truth tables and examples. Visit BYJU’S for more information.

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https://www.britannica.com/science/function-mathematics
SofiaAnneAranton_Function when used in Mathematics, is an expression or rule that shows the relationship between variables. Most functions are mainly used in mathematical formulas. Common examples of functions are polynomial function, trigonometric functions, and inverse functions.

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https://byjus.com/maths/business-mathematics/
SofiaAnneAranton_Business Mathematics mainly consist of business-related mathematical concepts since Mathematics is also the root of all businesses. It often deals with profit, loss, and interest. Business Mathematics is used to help record and manage records in businesses.

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Bonifacio Edclarence,
This video helps us understand what Logic is in math, it is always very important for us to find an overview or find out the introduction of each topic, because it is always the first step for us to understand the background of something very complex especially when it new to us

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INTRODUCTION to PROPOSITIONAL LOGIC - DISCRETE MATHEMATICS

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https://study.com/academy/lesson/critical-thinking-and-logic-in-mathematics.html
SofiaAnneAranton_Logic is the study on how to critically think statements that may be true or false. It is widely used in Mathematics in order to prove mathematical facts and theorems. It is the reason why we have our existing mathematical formulas because what we know today are results of their logical proofs.

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Logic means reasoning. The reasoning may be a legal opinion or mathematical confirmation and logic is the study of Major subareas including model theory, proof theory, set theory, and recursion theory. Logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power.

https://en.wikipedia.org/wiki/Mathematical_logic Christian Paul A. Torrico_Key Takeaway

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Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive powe…

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https://byjus.com/maths/business-mathematics/

This link provides a basic information about Mathematical Business, its Definitions and Formulas.

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IMPORTANT TOPICS COVERED IN BUSINESS MATHEMATICS 

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IMPORTANT TOPICS COVERED IN BUSINESS MATHEMATICS 

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https://www.google.com/url?sa=t&source=web&rct=j&url=https://byjus.com/maths/business-mathematics/&ved=2ahUKEwjp1byH7OD5AhV6gFYBHdLNA5QQFnoECAgQBQ&usg=AOvVaw2VUollMq775zwIFyETR8Lh

John Carlo G Paguntalan
•Basic math skills are those that involve making calculations of amounts, sizes or other measurements.They said that there are several topics in business mathematics, with examples being statistics, insurance, taxes, loans, and insurance.

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https://byjus.com/maths/mathematical-logic/

On this website, you will be provided other formulas and examples about Logic. This addresses the mathematical properties of formal systems of the topic such as their expressive or deductive power. Also, helps make you a more critical reader, listener, and thinker.

Domingo, Ma. Paulina Beatriz

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https://www.youtube.com/watch?v=-Pr3NoB9S9k&t=3s&ab_channel=WOWMATH

Sta. Clara, Rica Klarisse G.

Key Takeaways:
Ordered pair is the values x and y expressed between parentheses in an fixed order. The Relation is a set of ordered pairs. The first set of coordinates is referred to as the Domain, while the second set of coordinates is referred to as the Range. The concept of function refers to a connection in which each domain element corresponds to precisely one element of the range.

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https://www.youtube.com/watch?v=7yeGXgzJxbQ&ab_channel=DepEdTV-Official

Sta. Clara, Rica Klarisse G.

Key Takeaways:
Mathematical concepts like fractions, percentages, and decimals are explored in business mathematics, which is then used in the management of businesses and comprehending concepts connected to a business. Fractions, percentage, and decimals are all essential to business mathematics as it allows people to have the precise number when dealing with a business situation.

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https://www.google.com/url?sa=t&source=web&rct=j&url=https://en.m.wikipedia.org/wiki/Mathematical_logic%23:~:text%3DMathematical%2520logic%2520is%2520the%2520study,their%2520expressive%2520or%2520deductive%2520power.&ved=2ahUKEwi9zLuC7-D5AhUCp1YBHZn3BBIQFnoECAsQBQ&usg=AOvVaw25FYX-kTntAaQoWtaao0-F

John Carlo G Paguntalan
•Mathematical logic is the study of formal logic within mathematics. MATHEMACTICAL LOGIC is divided into four parts: model theory, proof theory, set theory, and recursion theory (also known as computability theory).

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https://www.youtube.com/watch?v=b5yjTKtr-Go&ab_channel=enginerdmath

Sta. Clara, Rica Klarisse G.

The field of study known as logic is concerned with investigating the truth or falsity of statements. To rephrase, logical reasoning may be used to determine the validity of a statement and draw conclusions about them.