Sets and Set Notation

Definition and Basic Concepts

A set is a collection of distinct objects, called elements. Sets are denoted by curly braces { } and named with capital letters. For example, S = {2, 5, 8}.

• Element notation: If 2 is an element of S, we write 2 ∈ S. If 9 is not an element, we write 9 ∉ S.

• Subset: A set A is a subset of B if every element of A is also in B. Notation: A ⊂ B or A ⊆ B.

Common sets of numbers:

• N: Natural numbers {1, 2, 3, ...}

• N0: Whole numbers {0, 1, 2, ...}

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

• Q: Rational numbers (can be written as p/q, q ≠ 0)

• I: Irrational numbers (decimals neither terminate nor repeat, e.g., , )

• R: Real numbers (all rational and irrational numbers)

Set Notation Types

• Roster notation: Listing elements, e.g., N = {1, 2, 3, ...}

• Set-builder notation: Describes properties, e.g., D = {x | x > 5 and x is even}

• Interval notation: Used for intervals of numbers, e.g., (-2, 9]

Interval Types, Notation, and Graphs

Intervals are used to describe subsets of real numbers. The table below summarizes the main types, their notation, and graphical representation.

Additional info: The image visually demonstrates how open and closed endpoints are represented on the number line, which is essential for understanding interval notation in algebra.

Properties of Real Numbers

Basic Properties

• Commutative property: ,

• Associative property: ,

• Additive identity:

• Additive inverse:

• Multiplicative identity:

• Multiplicative inverse: (for )

• Distributive property:

Example: illustrates the commutative property of multiplication.

Absolute Value and Distance

Absolute Value

The absolute value of a real number , denoted , is its distance from 0 on the number line.

• If ,

• If ,

Example: ,

Distance Between Two Points

The distance between real numbers and is .

• Example: Distance between and $4|-7 - 4| = |-11| = 11$

Exponents and Their Properties

Definition and Rules

For any positive integer , means multiplied by itself $n$ times.

• Product rule:

• Quotient rule: ,

• Power rule:

• Product to a power:

• Quotient to a power: ,

• Zero exponent:

Example:

Scientific Notation

Definition and Usage

Scientific notation expresses numbers as , where and is an integer.

• Example:

• Example:

Order of Operations

Rules

1. Calculate within grouping symbols first (parentheses, brackets).

2. Evaluate exponents.

3. Multiply and divide from left to right.

4. Add and subtract from left to right.

Example:

Polynomials and Operations

Definition and Structure

A polynomial in one variable is an expression of the form , where is a nonnegative integer and are real coefficients.

• Degree: The highest exponent of the variable.

• Leading coefficient: The coefficient of the term with the highest degree.

• Constant term: The term without a variable.

Example: has degree 7, leading coefficient 4, constant term -9.

Like Terms and Operations

• Like terms: Terms with the same variables raised to the same powers.

• Adding/Subtracting: Combine like terms.

Example:

Multiplying and Factoring Polynomials

Multiplication

• Use the distributive property:

• FOIL method for binomials: First, Outer, Inner, Last terms

Example:

Special Products

• Square of a sum:

• Square of a difference:

• Product of sum and difference:

Factoring Techniques

• Factor out common factors first.

• Factoring by grouping: Group terms to factor common binomials.

• Trinomials: Find two numbers whose product is and sum is .

• Special factorizations: Difference of squares, sum/difference of cubes.

Example:

Equations and Principles

Linear and Quadratic Equations

• Linear equation:

• Quadratic equation:

Solving Principles

• Addition Principle: If , then

• Multiplication Principle: If , then

• Zero Product Principle: If , then or

• Square Root Principle: If , then or

Example: Solve

Rational Expressions

Domain and Simplification

The domain of an expression is the set of real numbers for which it is defined. Division by zero is undefined.

• Example: Find the domain of Denominator factors: Domain: ,

Operations with Rational Expressions

• Simplify, multiply, divide, add, and subtract using common denominators.

• Complex rational expressions: Use LCD or reciprocal methods to simplify.

Radical Expressions

Roots and Properties

A number is an nth root of if . The symbol denotes the nth root.

• If is even,

• If is odd,

Example: , ,