6331306060_Natthaya by NATTHAYA THAKOENGSORAKHAN

The dimension of Matrix

6331306060 — 2021-09-08 01:22:43 UTC

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

Transpose

6331306060 — 2021-09-08 01:23:10 UTC

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

Addiction

6331306060 — 2021-09-08 01:23:44 UTC

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

Product/Multiplication

6331306060 — 2021-09-08 01:24:10 UTC

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_ax n_bwhen m_a is the number of rows of the matrix A and n_b is the number of columns of the matrix B
Warning: AB ≠ BA

Determinant

6331306060 — 2021-09-08 01:24:23 UTC

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

Inverse

6331306060 — 2021-09-08 01:24:39 UTC

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

What's Linear equations

6331306060 — 2021-09-08 01:25:48 UTC

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

Solving Linear equations by Matrix (Multiplication)

6331306060 — 2021-09-08 01:26:15 UTC

- 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.

What's set?

6331306060 — 2021-09-08 01:27:49 UTC

- 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.

Declaring set

6331306060 — 2021-09-08 01:28:36 UTC

> 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"

Set Notation

6331306060 — 2021-09-08 01:29:48 UTC

- 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.

Set equality

6331306060 — 2021-09-08 01:30:13 UTC

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

Venn Diagram

6331306060 — 2021-09-08 01:30:29 UTC

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

Operation of set

6331306060 — 2021-09-08 01:40:00 UTC

- 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.

Identity Matrix (I)

6331306060 — 2021-09-08 12:27:32 UTC

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.

Understand well

6331306060 — 2021-09-08 13:12:26 UTC

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

Still confusing

6331306060 — 2021-09-08 13:12:40 UTC

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

Understand well

6331306060 — 2021-09-08 13:19:48 UTC

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

Still confusing

6331306060 — 2021-09-08 13:20:00 UTC

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

Solving Linear equations by Matrix (Row Operation)

6331306060 — 2021-09-08 15:31:06 UTC

- 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

Subset

6331306060 — 2021-09-10 17:51:02 UTC

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

Universal set

6331306060 — 2021-09-10 18:08:15 UTC

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

Laws

6331306060 — 2021-09-10 18:11:37 UTC

Power set

6331306060 — 2021-09-10 18:12:10 UTC

- 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).

Cartesian product

6331306060 — 2021-09-10 18:15:53 UTC

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}

Still confusing

6331306060 — 2021-09-10 18:17:12 UTC

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

Understand well

6331306060 — 2021-09-10 18:57:38 UTC

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

Combination Operation

6331306060 — 2021-09-10 19:17:23 UTC

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

Proposition

6331306060 — 2021-09-11 06:29:54 UTC

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

Logical Operators: Negative Operator (Not)

6331306060 — 2021-09-11 06:33:39 UTC

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

Tautology

6331306060 — 2021-09-11 06:48:52 UTC

- A compound statement, that is always true

**not**
- P ∨ ∼P is tautology.

Logically equivalent

6331306060 — 2021-09-11 06:49:07 UTC

- 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.

Logical Operators: Conjunction (And)

6331306060 — 2021-09-11 06:56:21 UTC

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

Logical Operators: Disjunction (Or)

6331306060 — 2021-09-11 06:57:43 UTC

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

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

6331306060 — 2021-09-11 07:05:48 UTC

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

Logical Operators: Biconditional (If and only if)

6331306060 — 2021-09-11 07:05:54 UTC

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

Logically equivalent (Laws)

6331306060 — 2021-09-11 07:49:15 UTC

Truth Table

6331306060 — 2021-09-11 07:50:11 UTC

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

Gate Symbols

6331306060 — 2021-09-11 08:03:29 UTC

BOOLEAN ALGEBRA

6331306060 — 2021-09-11 08:07:14 UTC

HOW TO WRITE MULTI-INPUT

6331306060 — 2021-09-11 08:08:20 UTC

TRUTH TABLE EXPRESSION IN BOOLEAN

6331306060 — 2021-09-11 08:18:49 UTC

- Use binary number insted T and F

SIMPLIFICATION

6331306060 — 2021-09-11 08:20:06 UTC

BASIC PROPERTIES (LAWS)

6331306060 — 2021-09-11 08:21:35 UTC

What's Probability

6331306060 — 2021-10-04 01:11:05 UTC

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.

And Rule

6331306060 — 2021-10-04 01:11:49 UTC

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

Or Rule

6331306060 — 2021-10-04 01:12:00 UTC

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

Not Rule

6331306060 — 2021-10-04 01:12:17 UTC

- 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.

Combining Rule

6331306060 — 2021-10-04 01:12:24 UTC

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

Probability Basics

6331306060 — 2021-10-04 01:14:04 UTC

- 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

Conditional Probability

6331306060 — 2021-10-04 01:14:48 UTC

- We will have at least special conditional

Understand well

6331306060 — 2021-10-04 01:15:25 UTC

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

Still confusing

6331306060 — 2021-10-04 01:15:38 UTC

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

What's Combinatorics

6331306060 — 2021-10-04 04:29:59 UTC

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

Independent and Dependent Event

6331306060 — 2021-10-04 11:44:07 UTC

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

Mutually Exclusive Events

6331306060 — 2021-10-04 11:44:41 UTC

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.

Sum Rule

6331306060 — 2021-10-04 11:45:01 UTC

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

Product Rule

6331306060 — 2021-10-04 11:45:23 UTC

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

Permutation

6331306060 — 2021-10-04 11:46:09 UTC

- 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.

Combinations

6331306060 — 2021-10-04 11:48:08 UTC

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

Binomial Coefficients

6331306060 — 2021-10-04 11:49:54 UTC

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