Sets Overview

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