Slide 2-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

1 Copyright © 2005 Pearson Education, Inc.

SEVENTH EDITION and EXPANDED SEVENTH EDITION

Copyright © 2005 Pearson Education, Inc.

Chapter 2

Sets


2.1

Set Concepts


Set

A collection of objects, which are called elements or members of the set.

Listing the elements of a set inside a pair of braces, { }, is called roster form .


Well-defined Set

A set which has no question about what elements should be included.

Its elements can be clearly determined. No opinion is associated with the members.


Roster Form

This is the form of the set where the elements are all listed, each separated by commas.

Example: Set N is the set of all natural numbers less than or equal to 25.

Solution: N = {1, 2, 3, 4, 5,…25} The 25 after the ellipsis indicates that the elements continue up to and including the number 25.


Set-Builder (or Set-Generator) Notation

A formal statement that describes the members of a set is written between the braces.

A variable may represent any one of the members of the set.

Example: Write set B = {2, 4, 6, 8, 10} in set-builder notation.

Solution: 10{ and is an even number }.B x x N x


Finite Set

A set that contains no elements or the number of elements in the set is a natural number.

Example:

Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.


Infinite Set

An infinite set contains an indefinite (uncountable) number of elements.

The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.


Equal sets have the exact same elements in them, regardless of their order.

Symbol: A = B

Equal Sets


Cardinal Number

The number of elements in set A is its cardinal number.

Symbol: n(A)


Equivalent Sets

Equivalent sets have the same number of elements in them.

Symbol: n(A) = n(B)


Empty (or Null) Set

A null (or empty set ) contains absolutely NO elements.

Symbol: or


Universal Set

The universal set contains all of the possible elements which could be discusses in a particular problem.

Symbol: U


2.2

Subsets


Subsets

A set is a subset of a given set if and only if all elements of the subset are also elements of the given set.

Symbol:

To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B.


Example: Determine whether set A is a subset of set B.

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

Solution: All of the elements of set A are contained in set

B, so

Determining Subsets

A B .


Proper Subset

All subsets are proper subsets except the subset containing all of the given elements.

Symbol:


Determining Proper Subsets

Example: Determine whether set A is a proper subset of set B.

A = { dog, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B, and sets

A and B are not equal, therefore A B.


Determining Proper Subsets continued

Example: Determine whether set A is a proper subset of set B.

A = { dog, bird, fish, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B, but sets

A and B are equal, therefore A B.


Number of Distinct Subsets

The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.

Example: Determine the number of distinct subsets for

the given set { t , a , p , e }. List all the distinct subsets for the given set:

{ t , a , p , e }.


Solution: Since there are 4 elements in the given set, the

number of distinct subsets is

24 = 2 • 2 • 2 • 2 = 16 subsets. {t,a,p,e},

{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},

{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},

{t}, {a}, {p}, {e}, { }

Number of Distinct Subsets continued


2.3

Venn Diagrams and Set Operations


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 are divided into

subsets of U and are represented by circles.


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 of the two circles.

U

A B


Overlapping Sets

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

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

U

A B


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’


Intersection

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

Symbol: A B


Union

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

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

Symbol: A B


Subsets

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

Circle B is completely inside circle A.

,B AU

A

B


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.


2.4

Venn Diagrams with Three Sets


General Procedure for Constructing Venn Diagrams with Three Sets

Find the elements that are common to all three sets and place in region V.

U

A B

C

V I III

VII

VI IV

VIII

II


General Procedure for Constructing Venn Diagrams with Three Sets continued

Find the elements for region II. Find the elements in . The elements in this set belong in regions II and V. Place the elements in the set 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

A B

A B



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



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


Example: Constructing a Venn diagram for Three Sets Construct 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, {5}. A B C


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

and place in region II. {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 V.

{ 1, 2, 5, 8} Since 5 has been placed in region V, place 1 and 8 in

region IV.

A B

A C


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

VI. {5} There are now new numbers to be placed in this

region. 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. Using set C completes region VII.

B C


Example: Constructing a Venn diagram for Three Sets continued 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


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


2.5

Applications of Sets


Example: Toothpaste Taste Test

A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint.

In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types.

How many liked only the regular flavor? How many liked either one or the other or both? How many people did not like either flavor?


Solution

Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II.

35 people liked both flavors.

U

Regular Mint

35


Solution continued

Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint.

In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both.

74 – 35 = 39

62 – 35 = 27

U

Regular Mint

35 both

39 regular only

27 mintonly


Solution continued

“One or the other or both” represents the UNION of the two sets.

Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.


Solution continued

Take the total number of people in the entire sample and subtract the number who liked one or the other or both.

19 people did not like either flavor.

U

Regular Mint

35 both

62-35=27 Liked mint only

74-35=39 Liked mint only

19 liked neither


2.6

Infinite Sets


Infinite Sets

An infinite set is a set that can be placed in a one-to-one correspondence with a proper subset of itself.

These sets are “unbounded”.


Example: The Set of Multiples of Four

Show that it is an infinite set. {4, 8, 12, 16, 20, …,4n, …}Solution: We establish one-to-one

correspondence between the counting numbers and a proper subset of itself.

Given set: {4, 8, 12, 16, 20, …, 4n, …}

Proper subset: {4, 8, 12, 16, 20, …, 4n + 4, …}Therefore, the given set is infinite.


Countable Sets

A set is countable if it is finite or if it can be placed in a one-to-one correspondence with the set of counting numbers.

Any set that can be placed in a one-to-one correspondence with a set of counting numbers has cardinality aleph-null and is countable.

Slide 2-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

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Transcript of Slide 2-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.