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A declarative sentence with a definite meaning, having a truth value that’s either true (T) or false (F)

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- Help to find the truth values.
- Calculate all truth values by 2 n when n is the number of propositions in the statement.

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Logical Operators: Negative Operator (Not)

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Logical Operators: Negative Operator (Not)

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Changing the truth value of proposition by ' ~ ' or '  ¬ ' symbol.

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- Symbol: ' ^ '
- Truth value: If one of the propositions is false, got false.

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- Symbol: ' V '
- Truth value: If one of the propositions is true, got true.

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Logical Operators: Conditional (If..., then...)

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Logical Operators: Conditional (If..., then...)

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- Symbol: ' -> ' (one side arrow)
- Truth value: False when 'If (T), then (F)'.

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Logical Operators: Biconditional (If and only if)

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Logical Operators: Biconditional (If and only if)

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- Symbol: ' <-> ' (two sides arrow)
- Truth value: If all propositions have the same true value, got true.

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-  A compound statement, that is always true

**not**
- P ∨ ∼P is tautology.

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- Symbol: ' ≡ '
- Two compound statements that have the same all truth value which case-by-case.

**note**
- If a compound statement is a tautology, that compound statement is logically equivalent.

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The field of mathematics concerned with problems of selection, arrangement, and operation within a finite or discrete system.

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- Use binary number insted T and F

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- What's a proposition
- Logical Operators
- Tautology
- Logically equivalent

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Write as mxn when m is the number of row of matrix and n is the number of column of matrix

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A square matrix that, when multiplied by another matrix, equals that same matrix. (AI = A)
The elements on the principal diagonal are 1 and all other elements are 0.

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The two matrixes must have the same dimension
Solving: one by one at the same position.

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The number of rows of the second matrix must equal the number of columns of the first matrix
Write as AB when the first matrix is matrix A and the second matrix is matrix B
The dimension of the product is m a x n b when ma is the number of rows of the matrix A and nb  is the number of columns of the matrix B
Warning: AB ≠ BA

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Switching the rows with the columns.
We denote the transpose of matrix A by AT

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> Determinant must be the number, not matrix
> Write as |a| or det(A) for matrix A
> Tips for Thai: คูณลงลบคูณขึ้น

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Write the inverse of matrix A as A-1
How to check your answer: AA-1  = I

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-  A set is a collection of objects, things or symbols which are clearly defined.
-  The individual objects in a set are called the members or elements of the set.
-  A set must be properly defined so that we can find out whether an object is a member of the set.

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> Structure: Name = {Member(s) / Element(s)}
> Listing: A = {1,2,3}
> Describing: A = {x: x is an integer, 0 < x < 4}
> Telling from elements:  write "x is an element of set A" as "x ∈ A" and "x is not an element of set A" as "x  ∉  A"

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-  Finite set:  Finite sets are sets that have a finite number of members.

**note**  The number of elements in a finite set A is denoted by n(A).

-  Infinite set:  It is not possible to explicitly list out all the elements of an infinite set.
-  Empty set or Null Set: no elements in this set and represented by the symbol { } or Ø.

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Two sets are equal if they have the same elements.
Write as A = B when set A is equal to set B

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Create 1 circle or oval for 1 set. Put the set's name around the shape and labeled the elements within the shape.

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If every element of a set B is also a member of a set A, then we say “B is a subset of A”. We use the symbol ⊂ to mean “is a subset of” and ⊄ to mean “is not a subset of”

**note**
- Every set is a subset of itself i.e. for any set A, A ⊂ A
- The empty set is a subset of every set.
- For any two sets A and B, if A ⊂ B and B ⊂ A then A = B
-  The number of subsets for a finite set A is given by the formula: Number of subsets = 2 n(A) where n(A) = number of elements in the finite set A

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- Given the set S, the power set of S is the set of all subsets. The power of set S is denoted by P(S).

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The set of all elements under consideration, denoted by capital U or sometimes capital E.

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- Union(∪): The set of elements, which are in one set or in all.
- Intersection(∩):  The set of elements that are common to every set.
- Complement('): the set of elements that are members of the universal set U but not members of the set which the symbol.
- Set difference(-): All elements of the first set which are not also elements of the second set.

**note**
-  If A ⊂ B then A ∪ B = B.
-  If A ⊂ B then A ∩ B = A.

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Sample equations write as Ax+By = C when A, B, and C are numbers.
No power and No multiple

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Involving the intersection, union, and complement of sets.
Performing: the operations within brackets first. Other operations are performed from left to right.

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Solving Linear equations by Matrix (Multiplication)

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Solving Linear equations by Matrix (Multiplication)

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- Create matrix A, B, and X from linear equations structure (Ax+By = C)
- We can write the linear equations in the matrix system as AX = B then multiple A⁻¹ to this system we'll get A⁻¹AX = A⁻¹B. As we know A⁻¹A = I, so we get AX = B.
- Find A⁻¹
- Find X, 2 matrixes that equal the element at the same position will also equal.

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Solving Linear equations by Matrix (Row Operation)

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Solving Linear equations by Matrix (Row Operation)

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- Create a matrix with the coefficient of x and y on the left and the constant on the right divide them by the line
- Using any operation (addition or multiplication) with the rows of matrix to make the left side be the identity matrix
- The answer of equations is on the right side

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Let A and B are non-empty sets, their Cartesian product (denoted by A x B) is the set of all ordered pairs (a,b).
A × B = {(x, y) : x ∈ A, y ∈ B}

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-Addiction
-Multiplication
-Transpose
-Determinant
-How to solve linear eq. by matrix

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- How to find adj. for 3x3 dimension matrix
- Row Operation

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- Set Notation
- Declaring set
- Set equality
- Venn Diagram
- Operation of set

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- Boolean
- Logic Gate (when it's very complex)

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-  Creating the Power Set, write down the sequence of binary numbers.

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Independent means the outcome of one event does not affect the outcome of the other event.

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If E1 and E2 are mutually exclusive events, then E1 and E2 will not happen together. So the probability of the 2 events will be zero.

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If two events are mutually exclusive, that is, they cannot be done at the same time , then we must apply the sum rule.

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If two events are not mutually exclusive (that is, we do them separately), then we apply the product rule.

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- A permutation of a set S of objects is an ordered arrangement of the elements of S where each element appears only once.
- An ordered arrangement of r distinct elements of S is called an r-permutation.

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- Where as permutations consider order, combinations are used when order does not matter.

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They are the coefficients in the expansion of the expression (multivariate polynomial), (x + y )n . A binomial is a sum of two terms.

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The chance that a given event will occur.

**note**
- Probability can't be more than 1 and less than 0
- The total probability of all possible events is always 1.

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- Experiment: Activity with an 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: the relative likelihood that it will occur when you do the experiment

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- 2 independent events happen at the same time.
- We can find the probability of them from the product of their individual probabilities.

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- Either one of 2 different events will occur.
- We can find the probability of them from the sum of their individual probabilities.

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- When the probability of the event that we focus on is too much from the probability of the event that we didn't focus on.
-  We can find the probability from 1 minus the chance of it happening.

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- Combining the 'And Rule' and 'Or rule'

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- We will have at least special conditional

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- And Rule
- Or Rule
- Not Rule
- Combining Rule

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- Calculating by variable (I confuse the meaning of variable)