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2012 Pearson Prentice Hall. All rights reserved.

CHAPTER 2

Set Theory

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2012 Pearson Prentice Hall. All rights reserved. 2

2.3

Venn Diagrams and Set Operation

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Objectives1. Understand the meaning of a universal set.

2. Understand the basic ideas of a Venn diagram.

3. Use Venn diagrams to visualize relationships between two sets.

4. Find the complement of a set5. Find the intersection of two sets.

6. Find the union of two sets.

7. Perform operations with sets.

8. Determine sets involving set operations from a Venn diagram.9. Understand the meaning ofandand or.

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Universal Sets and Venn Diagrams

The universal set is a general set thatcontains all elements under discussion.

John Venn (18431923) created Venndiagrams to show the visual relationship among sets.

Universal set is represented by a rectangle

Subsets within the universal set are depicted by circles,

or sometimes ovals or other shapes.4

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Example 1: Determining Sets From aVenn Diagram

Use the Venn diagram todetermine each of the following

sets:

a. U

U= {, , $, M, 5}

b. A

A = {, }

c. The set of elements in Uthat are

not inA.

{$, M, 5}

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Representing Two Sets in a VennDiagram

Disjoint Sets:Two sets that have no

elements in common.

Proper Subsets:

All elements of setA

are elements of setB.

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Representing Two Sets in a VennDiagram

Equal Sets: IfA =B thenAB andB A.

Sets with Some Common

Elements

If setA and setB have at

least one element incommon, then the circles

representing the sets must

overlap.7

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Example 2: Determining sets from a VennDiagram

Solutions:

a. U= {a, b, c, d, e, f, g}

b. B = {d, e}c. {a, b, c}

d. {a, b, c, f, g}

e. {d}

Use the Venn Diagram todetermine:

a. U

b. B

c. the set of elements inA

but notB

d. the set of elements in U

that are not inB

e. the set of elements in

bothA andB.

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The Complement of a Set

The shaded region represents

the complement of setA or A'.This region lies outside the

circle.

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Example 3: Finding a Sets Complement

Let U= { 1, 2, 3, 4, 5, 6, 7, 8, 9}andA = {1, 3, 4, 7 }. FindA'.

Solution:

SetA'contains all the elements of

set U that are not in setA.

Because setA contains the

elements 1,3,4, and 7, these

elements cannot be members of

setA':

A'= {2, 5, 6, 8, 9}.

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The Intersection of Sets

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Example 4: Finding the Intersection ofTwo Sets

Find each of the following intersections:a. {7, 8, 9, 10, 11} {6, 8, 10, 12}

{8, 10}

b. {1, 3, 5, 7, 9} {2, 4, 6, 8}

c. {1, 3, 5, 7, 9}

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The Union of Sets

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E l 5 Fi di th U i f T

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Example 5: Finding the Union of TwoSets

Find each of the following unions:

a. {7, 8, 9, 10, 11} {6, 8, 10, 12}

b. {1, 3, 5, 7, 9} {2, 4, 6, 8}

c. {1, 3, 5, 7, 9}

a. {6, 7, 8, 9, 10, 11, 12}

b. {1, 2, 3, 4, 5, 6, 7, 8, 9}

c. {1, 3, 5, 7, 9}

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The Empty Set in Intersection and Union

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Example 6: Performing Set Operations

a. (A UB)'

Solution:AUB = {1, 3, 7, 8, 9, 10}

(AUB) = {2, 4, 5, 6}

b. A'B'

Solution

A'= {2, 4, 5, 6, 8, 10}

B' = {1, 2, 4, 5, 6, 9}

A'B'= {2, 4, 5, 6}

Always perform any operations inside parentheses first!Given:

U= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = { 1, 3, 7, 9 }

B = { 3, 7, 8, 10 }find

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E l 7 D t i i S t f

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Example 7: Determining Sets from aVenn Diagram

Use the diagram to determine each of the following sets:

a. AUB b. (AUB)' c. AB

d. (AB)' e. A' B f. AUB'

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E l 7 D t i i S t f

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Example 7: Determining Sets from aVenn Diagram

Solution

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Sets and Precise Use of Everyday

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Sets and Precise Use of EverydayEnglish

Set operations and Venn diagrams provide preciseways of organizing, classifying, and describing the

vast array of sets and subsets we encounter every

day.

Or refers to the union of sets

And refers to the intersection of sets

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