Section 2.1: Set Theory

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Learning Objectives

• Define sets
• Write sets three different ways
• Define the empty set
• Find the cardinality of a set
• Classify sets as finite or infinite
• Decide if two sets are equal or equivalent

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Introduction

Imagine organizing your music collection. You might group songs by genre such as rock, jazz, or classical, by decade such as the 80's, 90's, or 2000's, or by mood such as study, workout, or relaxation. In mathematics, we do something very similar. We organize objects into groups called sets.

Set theory is the branch of mathematics that studies these collections and the relationships between them. It provides a precise language for describing groups, connections, and patterns, and it serves as a foundation for much of modern mathematics. One reason it is important is that it allows us to communicate with precision. Instead of vague terms like many or most, we can describe exactly which elements belong to a group and how groups relate to one another.

Set theory also helps reveal connections between problems that may appear unrelated. Questions about overlapping classes, probability, or number systems often share the same underlying structure. By using sets, we can recognize and analyze these similarities more clearly. At the same time, studying sets develops important critical thinking skills such as classifying information, analyzing relationships, and reasoning carefully about inclusion and overlap.

These ideas appear in many real world contexts. Search engines use set operations to find relevant results, businesses analyze overlapping customer groups, social networks suggest connections based on shared relationships, and medical professionals identify patterns by examining combinations of symptoms.

At its core, a set is a well defined collection of distinct objects called elements. Although this definition may sound abstract, sets are everywhere, including the students in a class, the books on a shelf, or the courses you are taking this semester. As you begin studying set theory, you will see that it is more than just a mathematical topic. It is a powerful way to organize information and think clearly about the world.

Definition: Set

A set is a well-defined collection of distinct objects. The objects in a set can be anything: numbers, letters, people, books, or even other sets. Sets are typically denoted by capital letters A, B, C , etc. and the objects are listed within braces {  }.

Examples: 

• A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• B = {apple, banana, pear}

Definition: Epsilon, ϵ

The Greek letter, Epsilon (∈), is the mathematical symbol meaning "is an element of" or "belongs to." Its opposite is ∉, meaning "is not an element of."

Examples:

• 5 ∈ A  (read as "5 is an element of A")
• 4 ∉ A  (read as "4 is not an element of A")

Example #2.1.1 🤔

Let A = {1, 2, 3, 4, 5}, B = {1, 2, 3, 4, 5, …}, and C = all land animals. Decide whether each of the statements is true or false.

1. 1 ∈ A 
2. 1 ∈ B
3. 6 ∈ A
4. 6 ∉ B
5. 10,000 ∈ B
6. cat ∈ C
7. fish ∈ C
8. whale ∉ C
9. dog ∉ C
10. 7 ∈ C

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✅ Solution:

1. True
2. True
3. False
4. False
5. True
6. True
7. False
8. True
9. False
10. False

Section 2.1: Set Theory [In-Class Exercises]

1. Let A = the set of U.S. states, B = {5, 10, 15, 20, 25, ...}, and C = the set of U.S. Presidents. Decide whether each of the statements is true or false.
   1. 15 ∈ B 
   2. California ∉ A
   3. 30 ∉ B
   4. George Washington ∈ C
   5. Utah ∈ A
   6. 100 ∈ B
   7. 51 ∈ B
   8. Miami ∈ A
   9. 25 ∈ C
   10. Abraham Lincoln ∈ A
   11. 0 ∉ B
   12. Taylor Swift ∈ C

Answers

1. True
2. False
3. False
4. True
5. True
6. True
7. False
8. False
9. False
10. False
11. True
12. False

Definition: Finite Set

A finite set is a set that contains a specific, countable number of elements. You can list all elements and determine the exact size of the set.

Examples:

• {a, b, c, d} has 4 elements
• The set of students in this classroom

Definition: Infinite Set

An infinite set is a set that contains an unlimited number of elements. The elements continue without end.

Example:

• The set of all natural numbers {1, 2, 3, 4, ...}
• The set of all points on a line

Definition: Variable

A variable is a symbol (usually a letter like x, y, or n) that represents an unknown number or a quantity that can change.

Three Ways to Represent Sets

1. List/Roster Method
   • Lists all elements explicitly within braces.
   • Format: {element #1, element #2, element #3, .....}
   • Examples:
     • {1, 2, 3, 4, 5}
     • {red, blue, green, yellow}
     • {2, 4, 6, 8, .....}
2. Descriptive Method
   • Uses words to describe the set without mathematical notation.
   • Examples:
     • "The set of all positive integers less than 6"
     • "The set of primary colors"
     • "The set of all US state capitals"
3. Set Builder Notation
   • Uses mathematical notation with conditions/rules to define set membership.
   • Format: {x | condition(s)}
   • Examples:
     • {x | x ≥ 5} = "the set of numbers greater than or equal to 5"
     • {x ∈ ℕ | x < 10} = "the set of natural numbers less than 10"
     • {x | x = 2n, n ∈ ℤ} = "the set of all even integers"

Note: The symbols above ℕ and ℤ are called blackboard bold style letters, because mathematicians writing on blackboards (chalk boards) would trace over letters multiple times to make them stand out, creating a "bold" effect. These symbols/letters became standard for describing particular number sets. Here are most of the symbols used in that style and their meanings:

• The symbol ℕ represents the set of all natural numbers or {1, 2, 3, 4, ...}.
• The symbol 𝕎 represents the set of all whole numbers or {0, 1, 2, 3, 4, ...}
  $\mathbb{W}$
   
• The symbol ℤ represents the set of all integers or {..., −3, −2, −1, 0, 1, 2, 3, ...}.
• The symbol ℚ represents the set of all rational numbers.
• The symbol ℝ represents the set of all real numbers.
• The symbol ℂ represents the set of all complex numbers.

(For this text, we will only be working with ℕ, the set of all natural numbers)

When To Use Each M ethod

Method	Best for	Limitations
List/Roster	Small finite sets, concrete examples	Impractical for large/infinite sets
Descriptive	General communication, non-mathematical contexts	Can be ambiguous
Set-Builder	Mathematical sets, infinite sets, sets with clear rules	Requires mathematical background

Same Set, Three Different Ways

Here are the three different methods all representing the same exact set:

• Roster: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set-Builder: {x ∈ ℕ | 1 ≤ x ≤ 9}
• Descriptive: "The set of natural numbers from 1 to 9"

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Example #2.1.2 🤔

Rewrite each set the same in two other ways.

1. {1, 2, 3, 4, 5}
2. {x ∈ ℕ | 6 ≤ x ≤ 9}
3. "The set of natural numbers from 7 to 11"
4. {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

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✅ Solution:

1. Set-Builder: {x ∈ ℕ | 1 ≤ x ≤ 5}  and  Descriptive: "The set of natural numbers from 1 to 5"
2. Roster: {6, 7, 8, 9}  and  Descriptive: "The set of natural numbers from 6 to 9"
3. Roster: {7, 8, 9, 10, 11}  and  Set-Builder: {x ∈ ℕ | 7 ≤ x ≤ 11}
4. Set-Builder: {x ∈ ℕ | 1 ≤ x ≤ 19 and x is odd}  and  Descriptive: "The set of odd natural numbers from 1 to 19"

Section 2.1: Set Theory [In-Class Exercises]

2. Rewrite each set the same in two other ways.

1. "The set of natural numbers from 3 to 7"
2. {x ∈ ℕ | 5 ≤ x ≤ 10}
3. {2, 4, 6, 8, 10}
4. {x ∈ ℕ | 1 ≤ x ≤ 11 and x is odd}

Answers

1. Roster: {3, 4, 5, 6, 7}  and  Set-Builder: {x ∈ ℕ | 3 ≤ x ≤ 7}
2. Roster: {5, 6, 7, 8, 9, 10}  and  Descriptive: "The set of natural numbers from 5 to 10"
3. Set-Builder: {x ∈ ℕ | 2 ≤ x ≤ 10 and x is even}  and  Descriptive: "The set of even natural numbers from 2 to 10"
4. Roster: {1, 3, 5, 7, 9, 11}  and  Descriptive: "The set of odd natural numbers from 1 to 11"

Definition: Empty Set (or Null Set)

An empty set (or Null Set) is a set with no elements denoted by {  } or the symbol Ø.

Definition: Cardinal Number (or Cardinality) of a Set

Cardinal numbers describe "how many" elements are in a set - they represent the size or cardinality of a set.

The notation for cardinality is |A| or n(A).

Examples:

• The set A = {a, b, c} has cardinal number 3. So, |A| = 3. (Alternate notaion: \(n(A)=3\))
• The set B = {red, blue} has cardinal number 2. So, |B| = 2. (Alternate notation: n(B) = 2)
• The set C = {  } has cardinal number 0. So, |C| = 0. (Alternate notation: n(C ) = 0)

Example #2.1.3 🤔

Find the cardinal number of each set.

1. A = {3, 6, 9, 12, 15, 18, 21}
2. {x | x is any month of the year}
3. E = Ø
4. The set of all U.S. states.

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✅ Solution:

1. |A| = 7
2. 12
3. |E| = 0
4. 50

Section 2.1: Set Theory [In-Class Exercises]

3. Find the cardinal number of each set.
   1. A = {1, 2, 3, 4, 5, 6, 7}
   2. {λ | λ is any letter of the English alphabet}
   3. Z = {0}
   4. Z = Ø
   5. C = {fork, knife, spoon}
   6. S = {Ã, Ä, Å, Á, Æ}
   7. A = {1, 2, 3, 4, 5, ... , 100}

Answers

1. 7
2. 26
3. 1
4. 0
5. 3
6. 5
7. 100

Definition: Equivalent Sets

Two sets are equivalent if they contain exactly the same number of elements, regardless of what those elements actually are.

Examples:

• A = {1, 2, 3} and B = {red, blue, green} are equivalent (both have 3 elements)
• C = {a, b} and D = {x, y} are equivalent (both have 2 elements)

Definition: Equal Sets

Two sets are equal if they contain exactly the same elements. Every element in the first set must also be in the second set, and vice versa. Equal sets are denoted using the equality symbol (=).

Examples:

• A = {1, 2, 3} and B = {1, 2, 3} are equal sets, so A = B. 
• C = {a, b, c} and D = {c, a, b} are equal sets, so C = D.
• E = {apple, banana} and F = {apple, banana, pear} are not equal sets, so E ≠ F

Example #2.1.4 🤔

Determine whether each pair of sets is equal, equivalent, or neither.

1. {1, 16, 19, 91, 96}  and  {16, 96, 1, 91, 19}
2. {do, re, mi}  and  {baseball, football, basketball}
3. {four}  and  {f, o, u, r}
4. {x | x is any month of the year}  and  { y | y ∈ ℕ and y < 13}
5. {5, 7, 9, 11, 13}  and  {5, 7, 9, 11, 13, …}
6. {even natural numbers less than 10}  and  {2, 4, 6, 8}
7. {1, 2, 10, 20}  and  {2, 1, 20, 11}
8. {y | y is any odd natural number}  and  {1, 3, 5, 7, 9, ...}

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✅ Solution:

1. Equal; each set has 5 elements and the elements in both sets are the same.
2. Equivalent; each set has 3 elements, but the elements in both sets are not the same.
3. Neither; first set has 1 element and the second set has 4 elements; the elements in both sets are not the same.
4. Equivalent; each set has 12 elements, but the elements in both sets are not the same;   {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec}  &  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
5. Neither; first set has 5 elements and the second set is infinite; the elements in both sets are not the same.
6. Equal; each set has 4 elements and the elements in both sets are the same.
7. Equivalent; each set has 4 elements, but the elements in both sets are not the same.
8. Equal; each set has an infinite number of elements and the elements in both sets are the same.

Section 2.1: Set Theory [In-Class Exercises]

4. Determine whether each pair of sets is equal, equivalent, or neither.
   1. {1, 2, 3, 4, 5}  and  {6, 7, 8, 9, 10}
   2. {3, 7, 11, 13}  and  {13, 3, 7, 11}
   3. {5, 10, 15, 20, 25}  and  {5, 10, 15, 20, 25, …}
   4. {two}  and  {t, w, o}
   5. {odd natural numbers less than 10}  and  {1, 3, 5, 7, 9}
   6. {y | y is any even natural number}  and  {2, 4, 6, 8, 10, ...}
   7. {x | x is any day in the month of October}  and  { y | y ∈ ℕ and y < 32}
   8. {dog, cat, mouse}  and  {moose, fox, squirrel, pig}

Answers

1. equivalent
2. equal
3. neither
4. neither
5. equal
6. equal
7. equivalent
8. neither

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