Precalculus Foundations: Sets, Numbers, and Functions

Introduction to Sets and Numbers

Classification of Numbers

Understanding the different types of numbers and their relationships is foundational in mathematics. The main sets of numbers are:

• Natural Numbers (N): The counting numbers, N = {1, 2, 3, ...}

• Whole Numbers (N0): Natural numbers including zero, N0 = {0, 1, 2, 3, ...}

• Integers (Z): All positive and negative whole numbers, including zero, 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 (I): Numbers that cannot be written as a ratio of two integers (e.g., √2, π)

• Real Numbers (R): All rational and irrational numbers

Key Properties:

• Every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational and irrational number is a real number.

• Rational and irrational numbers are disjoint sets: Q ∩ I = ∅.

• The set of real numbers is the union of rational and irrational numbers: R = Q ∪ I.

Variables and Mathematical Objects

What is a Variable?

A variable is a symbol or object that represents elements of a certain set. For example, in mathematics, x can represent any real number, while in real life, a passport can represent a Canadian citizen.

Intervals and Inequalities

Interval Notation

Intervals are used to describe sets of numbers between two endpoints. The notation and corresponding set-builder forms are:

Interval	Set-Builder Notation
(a, b)	{x : a < x < b}
(a, b]	{x : a < x ≤ b}
[a, b)	{x : a ≤ x < b}
[a, b]	{x : a ≤ x ≤ b}
(a, ∞)	{x : x > a}
(−∞, b)	{x : x < b}
[a, ∞)	{x : x ≥ a}
(−∞, b]	{x : x ≤ b}
(−∞, ∞)	ℝ

Functions: Definitions and Examples

What is a Function?

A function is a rule that assigns to each element in a set A exactly one element in a set B. This concept is crucial for avoiding ambiguity in mathematical relationships.

• Example: Assigning each child to their unique age is a function, but assigning a person to a city (if there are multiple cities with the same name) may not be.

Function Notation and Representation

• Notation: If f is a function from set A to set B, we write f: A → B.

• Domain: The set A (inputs for the function)

• Range: The set of all possible outputs f(x) as x varies over the domain

• Example: f(x) = x^2 for x ∈ {−2, −1, 0, 1, 2}

Piecewise Defined Functions

Some functions are defined by different expressions over different parts of their domain. These are called piecewise defined functions.

• Example: The total annual fee for students in a science program depends on their residency status, with different formulas for Quebec residents, other Canadian residents, and international students.

Summary Table: Number Sets

Set	Definition	Example Elements
Natural Numbers (N)	Counting numbers	1, 2, 3, ...
Whole Numbers (N0)	Natural numbers + 0	0, 1, 2, ...
Integers (Z)	Positive and negative whole numbers	..., -2, -1, 0, 1, 2, ...
Rational Numbers (Q)	Numbers as fractions	1/2, -3/4, 5
Irrational Numbers (I)	Not expressible as fractions	√2, π
Real Numbers (R)	All rational and irrational numbers	All above