Topic 4 of 10

Sets

🎯 Learning Objectives

Understand the fundamental concept of sets as collections of objects
Master standard number set notations (ℕ, ℤ, ℚ, ℝ)
Apply subset relationships and visualize using Venn diagrams
Use set comprehension (set builder) notation effectively
Work with intervals for real numbers
Calculate cardinality for finite and infinite sets

5.1 Basic Definition

🔑 Definition: Set

Informal Definition: A set is a collection of items

Sets are fundamental mathematical objects that group related items together.

Examples of Sets

Days of week: {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
Prime numbers below 15: {2, 3, 5, 7, 11, 13}

5.2 Standard Number Sets

Number Set Hierarchy

ℕ: {0, 1, 2, 3, ...} ℤ: {..., -2, -1, 0, 1, 2, ...} ℚ: {p/q | p, q ∈ ℤ, q ≠ 0} ℝ: All rationals and irrationals

⚠️ Course Convention

Important: In this course, ℕ includes 0

ℕ = {0, 1, 2, 3, 4, ...}

Some textbooks exclude 0 from natural numbers, but we include it.

5.3 Subset Notation

🔑 Definition: Subset

A ⊆ B means "A is a subset of B"

Every element of A is also in B

Subset Examples

ℕ ⊆ ℤ (all natural numbers are integers)
ℤ ⊆ ℚ (all integers are rational)
ℚ ⊆ ℝ (all rationals are real)
Primes ⊆ ℕ (all primes are natural numbers)

Venn Diagram Representation

        ℝ (Real Numbers)
    ┌─────────────────────┐
    │      ℚ (Rationals)  │
    │  ┌──────────────┐   │
    │  │  ℤ (Integers)│   │
    │  │  ┌────────┐  │   │
    │  │  │ℕ(Naturals)│  │
    │  │  └────────┘  │   │
    │  └──────────────┘   │
    └─────────────────────┘

5.4 Set Comprehension (Set Builder Notation)

🔑 General Form

{x | x ∈ S, P(x)}

Read as: "the set of all x such that x belongs to S and x satisfies property P"

Components:

Generator: The underlying set (S)
Condition/Filter: Property P(x) that elements must satisfy

🏗️ Building Sets with Comprehension

Example 1: Even Natural Numbers

{n | n ∈ ℕ, n mod 2 = 0}

Or equivalently: {n | n ∈ ℕ, n is even}

Result: {0, 2, 4, 6, 8, 10, ...}

Example 2: Integers Between -6 and 6

{z | z ∈ ℤ, -6 ≤ z ≤ 6}

Result: {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Example 3: First 500 Natural Numbers

X = {n | n ∈ ℕ, n < 500}

Result: {0, 1, 2, ..., 499}

Example 4: Cubes of First 500 Natural Numbers

{n³ | n ∈ X} where X is defined above

Result: {0³, 1³, 2³, ..., 499³} = {0, 1, 8, 27, 64, ...}

💡 Flexibility in Notation

Can write conditions in various equivalent ways
Can use words or mathematical symbols
Precision matters, but readability is important

5.5 Intervals (for Real Numbers)

Types of Intervals

Closed Interval [a, b]:

Includes endpoints
[0, 1] = {r | r ∈ ℝ, 0 ≤ r ≤ 1}
Used for probabilities

Open Interval (a, b):

Excludes endpoints
(0, 1) = {r | r ∈ ℝ, 0 < r < 1}

Half-Open Intervals:

[a, b) = includes a, excludes b
(a, b] = excludes a, includes b

Visual Representation (Number Line)

Filled dot (●) means included

Open dot (○) means excluded

[2, 5]: ●────────● (includes both 2 and 5)

(2, 5): ○────────○ (excludes both 2 and 5)

[2, 5): ●────────○ (includes 2, excludes 5)

(2, 5]: ○────────● (excludes 2, includes 5)

5.6 Cardinality

🔑 Definition: Cardinality

The cardinality of a set is the number of elements in it

Notation: |A| represents the cardinality of set A

For Finite Sets

Simply count the elements:

|{1, 2, 3, 4, 5}| = 5
|{a, b, c}| = 3
|{factors of 24}| = |{1, 2, 3, 4, 6, 8, 12, 24}| = 8

For Infinite Sets

Use bijections (covered in later topics)

This is how we compare the "sizes" of infinite sets like ℕ, ℤ, ℚ, and ℝ

🧮 Practice Problems

Problem 1: Set Builder Notation

Write the following sets using set comprehension notation:

All odd natural numbers less than 20
All perfect squares between 1 and 100
All negative integers greater than -10

Solutions:

{n | n ∈ ℕ, n < 20, n is odd} = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
{n² | n ∈ ℕ, 1 ≤ n² ≤ 100} = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
{z | z ∈ ℤ, -10 < z < 0} = {-9, -8, -7, -6, -5, -4, -3, -2, -1}

Problem 2: Subset Relations

Determine if the following statements are true or false:

ℕ ⊆ ℝ
{2, 4, 6} ⊆ {even natural numbers}
{-1, 0, 1} ⊆ ℕ

Solutions:

True - All natural numbers are real numbers
True - 2, 4, 6 are all even natural numbers
False - -1 is not a natural number (ℕ = {0, 1, 2, 3, ...})

Problem 3: Intervals

Express the following intervals using set comprehension:

The closed interval [-3, 7]
The half-open interval (0, 1]
All real numbers greater than 5

Solutions:

{r | r ∈ ℝ, -3 ≤ r ≤ 7}
{r | r ∈ ℝ, 0 < r ≤ 1}
{r | r ∈ ℝ, r > 5} or using interval notation: (5, ∞)

Problem 4: Cardinality

Find the cardinality of the following sets:

A = {n | n ∈ ℕ, n² < 50}
B = {factors of 36}
C = {prime numbers less than 30}

Solutions:

|A| = 8
A = {0, 1, 2, 3, 4, 5, 6, 7} since 7² = 49 < 50 but 8² = 64 > 50
|B| = 9
B = {1, 2, 3, 4, 6, 9, 12, 18, 36}
|C| = 10
C = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}