# data[].attributes.body
The complement of set A is replaced by A'.
It's that the values in set A are not counted.

# data[].attributes.body
A matrix is a rectangular array of numbers or mathematical structures that can be added and multiplied with numbers. A group of real numbers arranged in a row. Each row has the same number with small brackets ( ) or brackets [ ] enclosed.

# data[].attributes.body
Transpose of Matrix or Transpose is the matrix obtained by swapping members. From the main row and from main to row of the master matrix.

# data[].attributes.body
Matrix Addition

The matrix that can be added together is must have the same dimensions And adding will bring members of the same position to add together.

# data[].attributes.body
Matrix Product (or Matrix Multiplication)

Matrix Multiplication by a real number is multiplying a real number with a matrix. In this multiplication method, it can multiply that real number by the members positioned in the matrix.(must be multiplied by every position) And the matrix can be any number of dimensions.

Matrix Multiplication require the following:
1.) The number of digits of the front matrix must be equal to the number of rows of the rear matrix.
2.) The dimensions of the resulting matrix are equal to The number of rows of the front digit times the number of digits of the back.

# data[].attributes.body
Identity Matrix

Identity Matrix is a diagonal matrix with a number on the diagonal of 1, assuming that the diagonal is drawn from the top-left member to the bottom-right (downward) element, all remaining 0 members are represented by the symbol In or just I. The part in quantum mechanics is written in bold 1 instead.

# data[].attributes.body
Determinant of Matrix

Determinant of Matrix is a function that results in a scalar quantity. Which depends on the value of n in the n×n dimension of the square matrix A. The determinant is the scale factor of a volume when A is used as a linear transformation. Determinants are used in multilinear algebra and calculus, which are used for substitution rules (substitution rule) in some groups of variables.

# data[].attributes.body
Matrix Inverse

Matrix Inverse refers to the inverse of the multiplication of the matrix. The matrix to find the inverse must have a value that is not equal to zero.

# data[].attributes.body
Linear Equations is an equation in which each term has only a constant or is the product of a constant and an exponential variable. Which will have the degree of the polynomial equal to 0 or 1.

# data[].attributes.body
Linear Expression is the command that performs the function of + - × ÷

# data[].attributes.body
1. Can't have exponents.

2. Cannot be multiplied or divided.

# data[].attributes.headline
Example of my skill in System of Linear Problem.

# data[].attributes.subject
Example of my skill in System of Linear Problem.

# data[].attributes.body
My problem solving of System of Linear Problem.

# data[].attributes.attachment_link.title
System of linear exercises - PATTARANIT SOOKKASEM

# data[].attributes.body
It is the study of mathematical structures which are characterized by specific values. Not continuous which does not require the concept of continuity. Most of the objects studied in this field are countable sets.

# data[].attributes.body
Set is a collection of objects put together the whole set.

# data[].attributes.attachment_caption
Example of variable declaration enclosed in curly braces.

# data[].attributes.body
Listing the elements

A variable declaration enclosed in curly braces.

Describing the elements

The set can be determined, if possible, by describe the composition.
Example
A = {x : x is an integer, x < 15 }
-----------------------------
∈ is a symbol of telling members.
∉ is a symbol of that's not a member.
Example
B = {1, 5, 6, 7, 8, 9,} then 1,5 ∈ B and 2,3 ∉ B
-----------------------------

# data[].attributes.body
Set Notation is a set by packing all members of the set are writing enclosed in curly braces. And between each member separated by a comma we can use three dots “…” instead of the remaining members that are not mentioned.
Example
C = {5, 10, 15, 20, 25, ... }
-----------------------------

# data[].attributes.body
Finite set is a set that can count all members and has a certain number.
Example
P = {5, 10, 15, 20, 25, ..., 100}
S = {x : x is an integer, 1 < x < 100}
-----------------------------
Infinite set is a set that cannot tell the number of members. Because it has a large number of members. The opposite side of Finite set.

Example
C = {3, 6, 9, ...}
A = {x : x is a triangle}
-----------------------------
N is the set of natural numbers
A is the set of fractions

# data[].attributes.body
Empty Set or Null Set is a set with no members and use the symbol
{ } or Ø .

# data[].attributes.body
Sets with the same number of members and the members of the set match up exactly one-to-one.
The symbol set A is equivalent to set B.
Example1
A = {w, x, y, z}
B = {5, 6, 7, 8}
Ans. A = B
-----------------------------
Example2
W = {2, 3, 4}
P = {5, 6, 7, 8}
Ans. W ≠ P
-----------------------------

# data[].attributes.body
Venn diagram is a set of diagrams that store the data in the set. It can be drawn as a circle or an oval.

# data[].attributes.body
Subset means 1 subset or subset, which is the part or group of things organized into parts or subgroups within a larger group.
⊂ is a symbol of “This is a subset of ...”.
⊄ is a symbol of that's not a subset.
Example1
A = {6, 7}
B = {5, 6, 7, 8}
Ans. A ⊂ B
-----------------------------
Example2
W = {2, 3, 4}
P = {5, 6, 7, 8}
Ans. W ⊄ P
-----------------------------
If the set is empty. It will use the following symbols: Ø ⊂ A
If both sets are subsets of each other. That means both sets are equal. A ⊂ B, B ⊂ A = A = B

# data[].attributes.body
Universal Set is a collection of all elements under consideration.
Denoted by an uppercase U or E.

# data[].attributes.body
Intersection is a set which consists of members that are members of a particular set and other set can be represented by symbols ∩ .
Such as A intersection set B is a set which consists of members of set A and set B. It can be represented by the symbol A ∩ B.
Example
A = {1, 2, 3, 4, 5}
B = {2,3}
Ans. A ∩ B = {2,3}
-----------------------------

# data[].attributes.body
Intersection of three sets are the value that all three sets have the same value same as the previous topic.
The method of writing will be added as A ∩ B ∩ C and split into A ∩ B, B ∩ C and A ∩ C.

# data[].attributes.body
Complement of the intersection of sets are take every values in the sets and universal. But not use intersect values within the sets. Write instead (A ∩ B)’ or (A ∩ B ∩ C)’ .

# data[].attributes.body
Union of sets are take all values in a set, such as set A and set B. If both sets are Union we use to take both sets.
Example
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {8, 9, 10}
Ans. A U B = {1, 2, 3, 4, 5, 8, 9, 10}
-----------------------------

# data[].attributes.body
It's the same as Complement of the intersection of sets, but change the Intersect to Union is written as (A U B)’ or (A U B U C)’ .

# data[].attributes.body
Combination Operation is to combine all the techniques from the past topics to apply the solution.

# data[].attributes.body
Drawing Venn Diagrams are draw circles (or ovals) to represent sets.

# data[].attributes.body
Definition if A and B are both subsets of set U, then the difference of sets A and B is a set consisting of members of set A but not members of B, denoted by A - B.

# data[].attributes.body
Distributive Laws

A∩(B∪C)=(A∩B)∪(A∩C)

A∩(B∪C)=(A∩B)∪(A∩C) and

A∪(B∩C)=(A∪B)∩(A∪C).

De Morgan's Laws

A∪B=B∪AA∪B=B∪A and

A∩B=B∩A

# data[].attributes.body
The set of all subsets of A. The power set of A is denoted by P(A). It might be confusing to read. It can denoted by P(S).

# data[].attributes.body
Cartesian Product of two Sets are mathematical operation which operates on multiple sets that results in sets (or sets of products). For sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) at a ∈ A and b ∈ B .

# data[].attributes.body
Logic is the subject of using rules reasoning

# data[].attributes.body
Because this matter will use facts as information. Which will represent data as (p, q, r, ...) is just an instruction. True facts are represented by T. False facts are represented by f.
---------------------
Negative Operator (NOT)
We use symbol is ~
Ex.(~p)

# data[].attributes.body
A proposition is a sentence that is only true or false. These sentences can be either in a affirmative or negative form. This principle requires a Truth Table. It can denoted by p ∧ q . The proposition is “p and q.”

# data[].attributes.body
Disjunction or "OR". It can denoted by p v q . The proposition is “p or q.”

# data[].attributes.body
Conditional Proposition or a Material Implication
It can denoted by p → q . The proposition is “ if p then q.”

# data[].attributes.headline
Biconditional Proposition or a Material Equivalence

# data[].attributes.subject
Biconditional Proposition or a Material Equivalence

# data[].attributes.body
Biconditional Proposition or a Material Equivalence
It can denoted by  p ↔ q . The proposition is “p if and only if q” .

# data[].attributes.body
Tautology is a fact proposition that is always true.

p∨ ∼ p is always true even when statement p is false or statement

∼ p is false.

# data[].attributes.body
Logically equivalent is a Compound propositions which have the same truth value in all possible cases are called logical equivalents. Compound propositions p and q are called logical equivalents.
- If p q is tautology
- The notation p ≡ q shows that p and q are logically equivalent.

# data[].attributes.body
Matrix

Transpose of Matrix

Matrix Arithmetic (Matrix Addition and Matrix Product (or Matrix Multiplication))

Set

Declaring Set

Set Notation

Finite set and Infinite set

Empty Set or Null SET

Set Equality

Venn Diagrams

Subset

Universal Set

Complement of a Set

Set Operations: Intersection

Complement of the intersection of sets

Union of sets

Complement of the union of sets

Drawing Venn Diagrams

Set Difference

Logic

# data[].attributes.body
Linear Equations

Intersection of three sets

Combination Operation of set topic

Important Properties

# data[].attributes.body
Pattaranit Sookkasem

# data[].attributes.body
Time/Timemachine

# data[].attributes.body
6331306069

# data[].attributes.body
Multimedia and animation technology

# data[].attributes.body
Mae Fah Luang University

# data[].attributes.body
Probability are measuring or estimating the possibility that how much will something happen or how true a statement will be? Probability ranges from 0 (0% chance or will not happen) to 100 (100% chance or will happen).
Statistics refers to numbers used to describe events or facts of a story that we want to study.

# data[].attributes.body
The probability of two independent events occurring as a result of the probabilities of each.
We called the AND rule because “this event happens AND that event happens”.
Example
Sarah has a bag containing 4 red balls and 5 green balls (9 balls in total). If she were to select a ball at random from the bag:

1. What is the probability that she selects a red ball?

2. What is the probability she selects a green ball?

Sol.

1. To solve this we realize that there are 4 red balls, out of 9 balls in total, leaving us with a probability of 4/9.

2. There are 5 green balls out of a total of 9 balls so the probability is 5/9.

# data[].attributes.body
The probability that one of two different events will occur is the sum of separate probabilities.
Example
Ashley is going to roll a dice. What is the probability he will get an odd number or a two?

Sol. Using the OR rule. P(odd or two) = P(odd) + P(two).
P(odd) = 3/6 and P(two) = 1/6.
P(odd or two) = (1/6) + (3/6) = (4/6).

# data[].attributes.body
The chance that an event will not happen is 1 minus the chance that it will happen.
Example
A bag contains 12 counters of different colours: 5 red, 4 white and 3 black. Find the probability of not selecting a red counter. The probability of selecting a red counter is 5/12, so the probability of not selecting a red counter is 1 - (5/12) which is (12/12) - (5/12) =7/12
A spinner is marked with the numbers 1, 2, 3 and 4. Find the probability, p, of the spinner showing a 4.

Sol.

1. Using the OR rule. P(odd or two) = P(odd) + P(two).
P(odd) = 3/6 and P(two) = 1/6.
P(odd or two) = (1/6) + (3/6) = (4/6).

Probabilities of events add up to 1, so to find the probability of the spinner showing a 4, add up the remaining probabilities and subtract this from 1.
0.5 + 0.2 + 0.12 + 0.82
p = 1 - 0.82 = 0.18
Ans. So the probability of the spinner showing a 4 must be 0.18.

# data[].attributes.body
More complex situations involve combining AND and OR rules.
It's very important to keep track of the people involved leave them confused. This is where most people come from Probability problem.
Example
Suppose a teacher wants to know the probability that a single student in her class of 30 students is taking either Art or English. She asks the class to raise their hands if they are taking Art and counts 13 hands. Then she asks the class to raise their hands if they are taking English and counts 21 hands. The teacher then calculates
Sol. P(Art or English)= (13+21)/30 = 33/30
Ans. The teacher knows that this is wrong because probabilities must be between zero and one, inclusive. After thinking about it she remembers that nine students are taking both Art and English. These students raised their hands each time she counted, so the teacher counted them twice. When we calculate probabilities we have to be careful to count each outcome only once.

# data[].attributes.body
- Experiment – Activity with uncertain outcome.
- Sample space – Complete list of all possible individual outcomes of an experiment.
- Event – A subset of the sample space
- Probability of an event is the relative likelihood that it will occur when you do the experiment.

# data[].attributes.body
Information about the probability of an event occurring on an event affect the probability of another event.