Sets Day 2

Warning: All the Venn Diagram construction and pictures will be done during class and are not included in this presentation. If you missed class you should get class notes from another student.

Example. Let U be the set of natural numbers less than or equal to 10.

Let A={2,4,6} and B={1,2,3,4,5}.

(Note: I constructed this Venn Diagram during class and referred to it throughout this presentation.)

Venn Diagrams

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

A′ is read “A complement,” or “A prime.”

Definition of Set Complement

Example. Let U be the set of natural numbers less than or equal to 10.

Let A={2,4,6} and B={1,2,3,4,5}.

Find A′ and B′.

Set Complement

Answers:

A′ = {1,3,5,7,8,9,10}

B′ = {6,7,8,9,10}

The intersection of sets A and B, symbolized by A∩B, is the set containing all the elements that are common to both A and B.

A∩B is read “A intersect B,” or “A AND B.”

Definition of Set Intersection

Example. Let U be the set of natural numbers less than or equal to 10.

Let A={2,4,6} and B={1,2,3,4,5}.

Find A∩B.

Answer: A∩B = {2,4}

Set Intersection

Let U = {a, b, c, d}A = {a, c} B = {b, d}

1. Find A′.

2. Find U′.

3. Find φ′.

4. Find A∩B.

5. Find (A∩B)′.

Example

A′ = B

U′ = φ

φ′ = U

A∩B = φ

(A∩B)′= U

Answers:

The union of set A and set B, symbolized by A∪B, is the set containing all the elements that are members of set A or of set B (or of both sets).

A∪B is read “A union B,” or “A OR B.”

Definition of Set Union

Example. Let U be the set of natural numbers less than or equal to 10.

Let A={2,4,6} and B={1,2,3,4,5}.

Find A∪B.

Answer: A∪B = {1,2,3,4,5,6}

Set Union

Let U = {a, b, c, d}A = {a, c} B = {b, d}

1. Find A∪B.

2. Find (A∪B )′.

3. Find A′∩B.

4. Find (A′∩B)′.

Example

A∪B = U

(A∪B )′ = φ

A′∩B = B

(A′∩B)′ = A

Answers:

The relationship between sets A, B, A∪B, and A∩B is given by the union rule:

n(A∪B) = n(A) + n(B) – n(A∩B)

Union Rule for Sets

Why? (We used aVenn Diagram to show this.)

1. If n(A) = 5, n(B) = 8, and n(A∩B) = 2, find n(A∪B).

Answer: n(A∪B)=5+8-2=11

2. If n(A)=12, n(A∪B)=22, and n(A∩B)=10, find n(B).

Answer: 20

Two Possible Union Rule Test Problems:

The difference of two sets A and B, symbolized A-B, is the set of elements that belong to set A but not to set B.

Example. Let U ={1,2,3,…,10}, A={2,4,6} and B={1,2,3,4,5}. Find A-B and B-A.

A-B={6}B-A={1,3,5}

Difference of Two Sets