Chapter 2 Section 3 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 2

Chapter 2

Sets

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 3

WHAT YOU WILL LEARN• Venn diagrams

• Set operations such as complement, intersection, union, difference and Cartesian product

• Equality of sets

• Application of sets

• Infinite sets

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 4

Section 3

Venn Diagrams and Set Operations

Chapter 2 Section 3 - Slide 5Copyright © 2009 Pearson Education, Inc.

Venn Diagrams

A Venn diagram is a technique used for picturing set relationships.

A rectangle usually represents the universal set, U. The items inside the rectangle may be divided

into subsets of U and are represented by circles.

Chapter 2 Section 3 - Slide 6Copyright © 2009 Pearson Education, Inc.

Disjoint Sets

Two sets which have no elements in common are said to be disjoint.

The intersection of disjoint sets is the empty set. Disjoint sets A and B are drawn in this figure. There are no elements in common since there is no overlap-

ping area between the two circles.

U

A B

Chapter 2 Section 3 - Slide 7Copyright © 2009 Pearson Education, Inc.

Overlapping Sets

For sets A and B drawn in this figure, notice the overlapping area shared by the two circles.

This area represents the elements that are in the intersection of set A and set B.

U

A B

Chapter 2 Section 3 - Slide 8Copyright © 2009 Pearson Education, Inc.

Complement of a Set

The set known as the complement contains all the elements of the universal set which are not listed in the given subset.

Symbol: A´

Chapter 2 Section 3 - Slide 9Copyright © 2009 Pearson Education, Inc.

Intersection

The intersection of two given sets contains only those elements common to both of those sets.

Symbol: AB

Chapter 2 Section 3 - Slide 10Copyright © 2009 Pearson Education, Inc.

Union

The union of two given sets contains all of the elements for both of those sets.

The union “unites”, that is, it brings together everything into one set.

Symbol: AB

Chapter 2 Section 3 - Slide 11Copyright © 2009 Pearson Education, Inc.

Subsets

When every element of B is also an element of A.

Circle B is completely inside circle A.

U

A

B

B A,

Chapter 2 Section 3 - Slide 12Copyright © 2009 Pearson Education, Inc.

Equal Sets

U

A B

When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A.

Both sets are drawn as one circle.

Chapter 2 Section 3 - Slide 13Copyright © 2009 Pearson Education, Inc.

The Meaning of and and or

and is generally interpreted to mean intersection

A B = { x | x A and x B }

or is generally interpreted to mean union

A B = { x | x A or x B }

Chapter 2 Section 3 - Slide 14Copyright © 2009 Pearson Education, Inc.

The Relationship Between n(A B), n(A), n(B), n(A B)

To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements common to both sets.

n(A B) = n(A) + n(B) – n(A B)

Chapter 2 Section 3 - Slide 15Copyright © 2009 Pearson Education, Inc.

Difference of Two Sets

The difference of two sets A and B symbolized by A – B, is the set of elements that belong to set A but not to set B. Region 1 represents the difference of the two sets.

A B x | x A and x B

BAU

I II III

IV

Chapter 2 Section 3 - Slide 16Copyright © 2009 Pearson Education, Inc.

Cartesian Product

The Cartesian product of set A and set B, symbolized A B, and read “A cross B,” is the set of all possible ordered pairs of the form (a, b), where

a A and b B.

Select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element in set A.

Copyright © 2009 Pearson Education, Inc. Chapter 2 Section 3 - Slide 17

Section 4

Venn Diagrams with Three Sets

And

Verification of Equality of Sets

Chapter 2 Section 3 - Slide 18Copyright © 2009 Pearson Education, Inc.

General Procedure for Constructing Venn Diagrams with Three Sets

Determine the elements that are common to all three sets and place in region V,

A B C.

U

A B

C

V I III

VII

VI IV

VIII

II

Chapter 2 Section 3 - Slide 19Copyright © 2009 Pearson Education, Inc.

General Procedure for Constructing Venn Diagrams with Three Sets (continued)

Determine the elements for region II. Find the elements in A B. The elements in this set belong in regions II and V. Place the elements in the set A B that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner.

U

A B

C

V I III

VII

VI IV

VIII

II

Chapter 2 Section 3 - Slide 20Copyright © 2009 Pearson Education, Inc.

General Procedure for Constructing Venn Diagrams with Three Sets (continued)

Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner.

U

A B

C

V I III

VII

VI IV

VIII

II

Chapter 2 Section 3 - Slide 21Copyright © 2009 Pearson Education, Inc.

General Procedure for Constructing Venn Diagrams with Three Sets (continued)

Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII.

U

A B

C

V I III

VII

VI IV

VIII

II

Chapter 2 Section 3 - Slide 22Copyright © 2009 Pearson Education, Inc.

Example: Constructing a Venn diagram for Three SetsConstruct a Venn diagram illustrating the following

sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8}Solution: Find the intersection of all three sets and place

in region V, A B C = {5}.

Chapter 2 Section 3 - Slide 23Copyright © 2009 Pearson Education, Inc.

Example: Constructing a Venn diagram for Three Sets (continued) Determine the intersection of sets A and B.

A B = {2, 5}

Element 5 has already been placed in region V, so 2 must be placed in region II.

Now determine the numbers that go into region IV.

A C = {1, 5, 8}

Since 5 has been placed in region V, place 1 and 8 in region IV.

Chapter 2 Section 3 - Slide 24Copyright © 2009 Pearson Education, Inc.

Example: Constructing a Venn diagram for Three Sets (continued) Now determine the numbers that go in region

VI. B C = {5}

There are no new numbers to be placed in region VI.

Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III, in which we must write 4. Using set C completes region VII, in which we must write 3.

Chapter 2 Section 3 - Slide 25Copyright © 2009 Pearson Education, Inc.

Example: Constructing a Venn diagram for Three Sets (continued) Now place the

remaining elements in U (6 and 7) in region VIII.

The Venn diagram is then completed.

U

A B

C

V I III

VII

VI IV

VIII

II 2 4

51,8

3

6 7

Chapter 2 Section 3 - Slide 26Copyright © 2009 Pearson Education, Inc.

Verification of Equality of Sets

To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams.

If both statements represent the same regions of the Venn Diagram, then the statements are true for all sets A and B.

Chapter 2 Section 3 - Slide 27Copyright © 2009 Pearson Education, Inc.

Example: Equality of Sets

Determine whether (A B)´ = A´ B´ for all sets A and B.

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Solution

Draw a Venn diagram with two sets A and B and label each region.

BAU

I II III

IV

Find (A B)´. Find A´ B´.Set Regions

A I, II

B II, III

A B II

(A B)´ I, III, IV

Set Regions

A´ III, IV

B´ I, IV

A´ B´ I, III, IV

Chapter 2 Section 3 - Slide 29Copyright © 2009 Pearson Education, Inc.

Solution

Both statements are represented by the same regions, I, III, and IV, of the Venn diagram. Thus,

(A B)´ = A´ B´ for all sets A and B.

Set Regions

A I, II

B II, III

A B II

(A B)´ I, III, IV

Set Regions

A´ III, IV

B´ I, IV

A´ B´ I, III, IV

Chapter 2 Section 3 - Slide 30Copyright © 2009 Pearson Education, Inc.

De Morgan’s Laws

A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms.

(A B)´ = A´ B´

(A B)´ = A´ B´