Math Rules, Order of Operations and Sets

• The document provided consists of math notes and formulas.

Rules of Indicating Signs

• The sum of two positive numbers is positive.
• The sum of two negative numbers is negative.
• The sum of two numbers with different signs has the sign of the larger number.
• The product of two numbers with the same sign is positive.
• The product of two numbers with different signs is negative.
• The quotient of two numbers with the same sign is positive.
• The quotient of two numbers with different signs is negative.

Order of Operations

• Perform mathematical operations in the following order
• Calculate what is inside the parentheses first
• Calculate the values of exponents and roots.
• Perform multiplication and division operations.
• Perform addition and subtraction operations.

Definition of a Set

• A set is a collection of distinct objects, either tangible or abstract, that can be distinguished from each other by precise and consistent criteria.

Representing Sets and Elements

• Sets are denoted by uppercase letters like A, B, C, D, etc.
• Elements of a set are denoted by lowercase letters such as a, b, c, d, etc.
• In mathematical notation, a set is represented by enclosing its elements within curly braces { }.

Relationships Between Elements and Sets

• If an element x is a member of set A, mathematically expressed as x ∈ A
• If an element x is not a member of set A, mathematically expressed as A ∉ 8

Ways to Represent a Set

• Roster Method: Listing all elements, e.g., A = {c, a, r} for the letters in "car".
• Descriptive Method: Describing the set's properties, e.g., B = {x; x ∈ days of the week}
• Rule Method: Defining the set using a rule, e.g., C = {x; x ∈ N: 2 ≤ x > 8} for even numbers 2, 4, 6.

Subset Definition

• Set A is a subset of set B if every element of A is also an element of B, which is denoted as A ⊆ B.

Equal Sets Definition

• Sets A and B are equal (A = B) if A ⊆ B and B ⊆ A, meaning they contain the same elements.

Empty Set Definition

• The empty set is a set that contains no elements, symbolized as ( or {}.

Universal Set Definition

• The universal set (U) consists of all elements of the sets.

Union of Two Sets (U)

• Denoted as A U B and includes all unique elements from both sets A and B. Venn diagrams can represent it.

Intersection of Two Sets (∩)

• Denoted as A ∩ B and includes elements common to both sets A and B, also representable in Venn diagrams.

Difference Between Two Sets ( - )

• Denoted as A - B is the set of elements in A that are not in B.
• A - B is not the same B-A.

Symmetric Difference Between Two Sets ( ⊕ )

• Denoted as A ⊕ B, is the set of elements in A or B, but not in their intersection; and can be showed in Venn diagrams.

Complement of a Set ( A )

• The complement of set A ( A ) includes elements in the universal set U but not in A.

Numerical Sets - Natural Numbers

• Natural Numbers: N = {1, 2, 3, 4, ...}

Whole Numbers

• Whole Numbers: W = {0, 1, 2, 3, 4, ...}

Integers

• Integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers

• Rational Numbers: Q = {m/n : m ∈ Z, n ∈ Z, n ≠ 0}

Real Numbers

• Real Numbers Include all numbers and are represented on a number line.

Adding and Subtracting Rational Numbers

• If the fractions have the same denominator, keep the denominator and add or subtract the numerators
• Otherwise, you need to unify the denominators

Process of Multiplying Rational Numbers

• Multiply the numerators and denominators

Process of Dividing Fractions

• Convert the division to multiplication by inverting the second fraction.

Rounding Numbers

• If the digit after the rounding place value is less than 5, cut off these digits.
• If the digit is 5 or more, add 1.

Definition of Exponents

• Where m is the base and n is the exponent.

Rules of Exponents

• (aⁿ)ᵐ = aⁿˣᵐ
• (a × b)ⁿ = aⁿ × bⁿ
• aⁿ × aᵐ = aⁿ⁺ᵐ
• aⁿ/aᵐ = aⁿ⁻ᵐ
• (a/b)ⁿ = aⁿ/bⁿ
• a⁻ⁿ = 1/aⁿ
• (a/b)⁻ⁿ = (b/a)ⁿ
• a⁰ = 1

Definition of an Algebraic Term

• An algebraic term consists of a constant (number) or a variable, or a product of a constant and a variable.
• The exponent of the variable must be a non-negative integer such as 5, x^4, 2x^3, 2x^2y^4, -10.

Degree of Algebraic Term Notes

• The degree of an algebraic term is the sum of the exponents of its variables.
• The degree of a constant term is always zero.
• The coefficient is the number by which the variable is multiplied.

Polynomial Definition

• A polynomial is the total number of algebraic terms, such as 7 + 5x^2 - 4x^3 (43, 5x^2, and 7).

Writing Polynomials

• Polynomials are written from the largest Degree number
• The degree is determined by its largest Degree number

Like Terms Definition

• Like terms are terms with the same variable raised to the same power.

Performing Algebraic Operations

• Combining the coefficients of like terms. This involves adding only the coefficients of the similar numbers.
• Subtraction is achieved by subtracting coefficients. Remember to change minus signs to (+) and change the sign.
• Multiplication involves distributing each term and the like terms are combined.

Computing Value

• Calculate the polynomial by inserting the variables and numbers.

Factoring Polynomials

• Factoring involves a polynomial by writing it as a product. Use common factor or differences of squares when solving polynomials.

Algebraic Fractions

• Are fractions consisting of polynomial expressions in the numerator and denominator.
• Can be simplified by identifying common expressions.

Matrices

• Matrices - a set of real numbers organized on rows and columns
• To find the dimensions of the matrix, make sure the number of rows and columns is correct
• Matrix addition involves adding corresponding elements of same size matrices.,
• Multiply the corresponding elements and add. - It will often involve multiplying each element by its constant value.

Matrix

• The product of two matrices requires that the number of columns in the first matrix equals the number of rows in the second.

Transpose of a Matrix (At)

• Involves interchanging its rows into columns.

Matrix Inverse

• If a matrix has a matrix determinant that is not zero, the matrix has an inverse.
• The inverse is calculated by using basic calculation.

Linear Equations

• Are expressions with operations with different variables, that are the 1st Degree

Linear Equation Guidelines

• Consists of getting same terms to one side and numbers to the other to begin steps to solve the Linear equation
• The steps are add, substract, multiply or divide.
• The value of 𝒂 has to equal a true real Number.

The rules for problem solving

• Is for a single value and true value

Quadratic Equations.

• Are on the form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
• Can be solved with the overall solution/method/formula.

Determining With the Formula

• The formula has three different possible calculations, which are three rules.
• 1 If A is true, for the equation, there are two possible real and very distinctive real numbers.
• 2 . I fA is zero, the equation has two numbers that are related.
• 3. If A is negative, there has to be an applicable real value for solving the equation

Quadrilaterals

• Four sides
• Examples: Square, rectangle

Rectangles

• Four sides
• All are perpendicular
• Two sides both have to be in equal

All Calculations for a Rectangle

• Perimeter(P)
• Area(Area= L times W)

Square - all calculations

• Perimeter( P==side L)
• Area

Triangle

• Side +Side+Side
• Length- given with standard following formula
• Three Points= 180 degrees.

Circle

• Calculate the diameter; the circle has a certain area,
• Set amount of Points and Distance.
• Perimeter,Area