Sets Overview

Study Notes

Sets

A set is a well-defined collection of distinct objects.
Members of a set are called elements.
Sets are typically denoted by capital letters (e.g., A, B).
Elements are typically denoted by lowercase letters (e.g., a, b).
Set membership is denoted by the symbol ∈ (e.g., a ∈ A means "a is an element of A").
Set non-membership is denoted by the symbol ∉ (e.g., a ∉ A means "a is not an element of A").
Empty set (∅ or {}). A set with no elements.

Types of Sets

Finite sets: Sets with a limited number of elements.
Infinite sets: Sets with an unlimited number of elements.
Natural numbers (N): {1, 2, 3, ...}
Whole numbers (W): {0, 1, 2, 3, ...}
Integers (Z): {...-3, -2, -1, 0, 1, 2, 3,...}
Rational numbers (Q): Numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
Irrational numbers: Numbers that cannot be expressed as a fraction of two integers.
Real numbers (R): The set of all rational and irrational numbers.
Complex numbers (C): Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).

Set Operations

Union (∪): The union of two sets A and B (A ∪ B) contains all elements that are in A or in B or in both.
Intersection (∩): The intersection of two sets A and B (A ∩ B) contains all elements that are in both A and B.
Difference (∖): The difference of set A and set B (A ∖ B) contains all elements that are in A but not in B.
Complement (A'): The complement of set A (A') contains all elements in the universal set that are not in A.
Subset (⊆): If every element of set A is also an element of set B, then A is a subset of B (A ⊆ B).
Proper subset (⊂): If A is a subset of B and A is not equal to B, then A is a proper subset of B (A ⊂ B).

Number Systems

Natural numbers (ℕ): Counting numbers (1, 2, 3, ...)
Whole numbers (ℤ⁺): Non-negative integers (0, 1, 2, 3, ...)
Integers (ℤ): Whole numbers and their opposites (...-3, -2, -1, 0, 1, 2, 3, ...)
Rational numbers (ℚ): Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0
Irrational numbers (ℚ'): Numbers that cannot be expressed as a fraction of two integers (e.g., √2, π)
Real numbers (ℝ): The set of all rational and irrational numbers.
- Includes all points on a number line.
Imaginary numbers: Numbers of the form bi, where b is a real number and i is the imaginary unit.
Complex numbers (ℂ): Numbers of the form a + bi, where a and b are real numbers.
- Combining real and imaginary parts.

Basic Mathematical Operations

Addition (+)
Subtraction (-)
Multiplication (× or *)
Division (÷ or /)
Exponents (e.g., 2³ = 8)
Roots (e.g., √4 = 2)

Ordering Numbers

Comparing numbers using inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

Fundamental Properties of Operations

Commutative property: a + b = b + a, a × b = b × a
Associative property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
Distributive property: a × (b + c) = (a × b) + (a × c)

Basic Geometry

Points, lines, planes

Algebra

Solving equations

Statistics

Measures of central tendency (mean, median, mode)
Measures of dispersion (variance, standard deviation)
- Used to describe data sets.

Logic

Statements and connectives
Truth tables
Logical equivalencies

Functions

Domain and range
Types of functions (linear, quadratic, exponential, etc.)

Calculus

Limits
Derivatives
Integrals

Probability and Statistics

Basic concepts of probability
Discrete and continuous random variables
Distributions (normal, binomial, etc.)

Financial Mathematics

Simple and compound interest
Loans and mortgages
Annuities
Present value and future value

Description

This quiz covers the fundamental concepts of sets, including definitions, symbols, and types of sets. You'll learn about finite and infinite sets, as well as specific sets like natural, whole, rational, and irrational numbers. Test your understanding of set membership and notation!