Thinking Mathematically

Chapter 2 Set Theory2.1 Basic Set Concepts

Basic Set Concepts

•A set is a collection of objects. Each object is called an element of the set.

•A set must be well defined:

Its contents can be clearly determined

Its clear if an object is or is not a member of the set.

Representing SetsWord Description: Describe the set in

your own words, but be specific so the elements are clearly defined.

Roster Method: List each element, separated by commas, in braces.

Set-Builder Notation: {x | x is … word description}.

The Set of Natural Numbers

N = {1,2,3,4,5,…}

This is an example of a set We will be talking a lot more about sets of

numbers in Chapter 5

Examples: Representing SetsExercise Set 2.1 #3, 5, 13, 15, 25

• Well defined sets (T/F):– The five worst U.S. presidents

– The natural numbers greater than one million

• Write a description for the set{6, 7, 8, 9, …, 20}

• Express this set using the roster method:The set of four seasons in a year.

{x | x N and x > 5 }

The Empty Set

The empty set, also called the null set, is the set that contains no elements.

The empty set is represented by { } or Ø

Examples: Empty Sets

Exercise Set 2.1 #35, 37, 41, 45

Which sets are empty

• {x | x is a women who served as U.S. president before 2000}

• {x | x is the number of women who served as U.S. president before 2000}

• {x | x <2 and x > 5}

• {x | x is a number less that 2 or greater than 5}

The Notation and

The symbol is used to indicate that an object is an element of a set. The symbol is used to replace the words “is an element of”

The symbol is used to indicate that an object is not an element of a set. The symbol is used to replace the words “is not an element of”

Example: Set elements

Exercise Set 2.1 #51, 59, 63 (T/F)• 5 { 2, 4, 6, …, 20}

• 13 {x | x N and x < 13 }

• {3} {3, 4}

Definition of a Set’s Cardinal Number

The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. The symbol n(A) is read “n of A”.Repeated elements are not counted.

Exercise Set 2.1 #71

C = {x | x is a day of the week that begins with the letter A}

n( C) = ?

Definition of a Finite Set

Set A is a finite set if n(A) = 0 or n(A) is a natural number. A set that is not finite is called an infinite set.

Exercise Set 2.1 #91

{x | x N and x >= 100}

Finite or infinite?

Definition of Equality of Sets

Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B.

Definition of Equivalent Sets

Set A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets, n(A) = n(B).

Exercise Set 2.1 #85

A = { 1, 1, 1, 2, 2, 3, 4}

B = {4, 3, 2, 1}

Are these sets equal?

Are these sets equivalent?

Thinking Mathematically

Chapter 2 Set Theory2.3 Venn Diagrams and Set Operations

[we’ll come back to 2.2]

Definition of a Universal Set

A universal set, symbolized by U, is a set that contains all of the elements being considered in a given discussion or problem.

Exercise Set 2.3 #3

A = {Pepsi, Sprite}

B = {Coca Cola, Seven-Up}

Describe a universal set that includes all elements in sets A and B

Venn Diagrams

“Disjoint” sets have no elements in common.

All elements of B are also elements of A.

UBA

U A

B

The sets A and B have some common elements.

UA B

Definition of the Complement of a Set

The complement of set A, symbolized by A´, is the set of all elements in the universal set that are not in A.

This idea can be expressed in set-builder notation as follows:

A´ = {x | x U and x A }.

Complement of a Set

U

A

A’

Example: Set Complement

Exercise Set 2.3 #11U = {1, 2, 3,…, 20}

A = {1, 2, 3, 4, 5}

B = {6, 7, 8, 9}

C = {1, 3, 5, …, 19}

D = {2, 4, 6, …, 20}

C´ = ?

Definition of Intersection of Sets

The intersection of sets A and B, written AB, is the set of elements common to both set A and set B. This definition can be expressed in set builder notation as follows:

A B = { x | x A AND x B}U

A B

Definition of the Union of Sets

The union of sets A and B, written A B, is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set-builder notation as follows:

A B = {x | x A OR x B}U

A B

The Empty Set in Intersection and Union

For any set A:1. A ∩ = 2. A = A

Examples: Union / IntersectionExercise Set 2.3 #17, 19, 33, 35

U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}

• A B = ?• A B = ?• A = ?• A ∩ = ?

Cardinal Number of the Union of Two Sets

n(A U B) = n(A) + n(B) – n(A ∩B)

Exercise Set 2.3 #93– Set A 17 elements– Set B 20 elements– There are 6 elements common to the two sets– How many elements in the union?

Thinking Mathematically

Chapter 2 Set Theory2.2 Subsets

Definition of a Subset of a Set

Set B is a subset of set A, expressed asB A

if every element in set B is also an element in set A.

U A

B

Every set is a subset of itself: A A

Definition of a Proper Subset of a Set

Set B is a proper subset of set A, expressed as B A, if set B is a subset of set A and sets A and B are not equal ( A B ).

What is an improper subset?

The Empty Set as a Subset

1. For any set B, B.2. For any set B other than the empty set,

B.

Example: Subsets

• Exercise Set 2.2 #3, 45, 43, 47

• {-3, 0, 3} ____ {-3, -1, 1, 3}

• (, , both, neither)

• {Ralph} {Ralph, Alice, Trixie, Norton} (T/F)

• Ralph {Ralph, Alice, Trixie, Norton} (T/F)

{Archie, Edith, Mike, Gloria} (T/F)

Thinking Mathematically

Chapter 2 Set Theory2.4 Set Operations and Venn Diagrams

With Three Sets

• U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}

B = {1, 2, 3}

C = {2, 3, 4, 5, 6}

(A B) ∩ (A C)

• U = {a, b, c, d, e, f, g, h}A = {a, g, h}B = {b, h, h}C = {b, c, d, e, f}(A B) ∩ (A C)

Example: Operations with three sets Exercise Set 2.4 #3, 15

Example – Venn DiagramsExercise Set 2.4 #35, 37

A B

C

1, 2, 3 4,5 10, 11

7, 86

9

12

U

A B = ?

13

(A B)’ = ?

Example – Venn DiagramsExercise Set 2.4 #27, 29

A B

C

III III

IV VVI

VII

U

A C = ? A ∩ B = ?

De Morgan’s Laws(using Venn Diagrams as a proof)

• (A U B)' = A' ∩ B': The complement of the union of two sets is the intersection of the complement of those sets.

UA B U

A B

UA B

De Morgan’s Laws

• (A ∩ B)' = A' U B': The complement of the intersection of two sets is the union of the complement of those sets.

UA B

UA B

UA B

Examples: DeMorgan’s Laws

U = {1, 2, 3, 4, 5, 6, 7}

A = {1, 3, 5, 7}

B = {1, 2, 3}• (A ∩ B) ' = ?

• A ' U B ' = ?

Thinking Mathematically

Chapter 2 Set Theory