- By
**baris** - Follow User

- 185 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Chapter 2' - baris

Download Now**An Image/Link below is provided (as is) to download presentation**

Download Now

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Chapter 2: The Basic Concepts of Set Theory

Chapter 2: The Basic Concepts of Set Theory

2.1 Symbols and Terminology

2.2 Venn Diagrams and Subsets

2.3 Set Operations and Cartesian Products

2.4 Surveys and Cardinal Numbers

2012 Pearson Education, Inc.

Symbols and Terminology

- Designating Sets
- Sets of Numbers and Cardinality
- Finite and Infinite Sets
- Equality of Sets

2012 Pearson Education, Inc.

Designating Sets

A set is a collection of objects. The objects belonging to the set are called the elements, or members of the set.

Sets are designated using:

1) word description,

2) the listing method, and

3) set-builder notation.

2012 Pearson Education, Inc.

Designating Sets

Word description

The set of even counting numbers less than 10

The listing method

{2, 4, 6, 8}

Set-builder notation

{x|x is an even counting number less than 10}

2012 Pearson Education, Inc.

Designating Sets

Sets are commonly given names (capital letters).

A = {1, 2, 3, 4}

The set containing no elements is called the

empty set (null set) and denoted by { } or

To show 2 is an element of set A use the symbol

2012 Pearson Education, Inc.

Example: Listing Elements of Sets

Give a complete listing of all of the elements of the set {x|x is a natural number between 3 and 8}

Solution

{4, 5, 6, 7}

2012 Pearson Education, Inc.

Sets of Numbers

Natural (counting) {1, 2, 3, 4, …}

Whole numbers {0, 1, 2, 3, 4, …}

Integers {…,–3, –2, –1, 0, 1, 2, 3, …}

Rational numbers

May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333…

Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat.

Real numbers {x | x can be expressed as a decimal}

2012 Pearson Education, Inc.

Cardinality

The number of elements in a set is called the cardinal number, or cardinality of the set.

The symbol n(A), read “n of A,” represents the cardinal number of set A.

2012 Pearson Education, Inc.

Example: Cardinality

Find the cardinal number of each set.

a) K = {a, l, g, e, b, r}

b) M = {2}

c)

Solution

a) n(K) = 6

b) n(M) = 1

c)

2012 Pearson Education, Inc.

Finite and Infinite Sets

If the cardinal number of a set is a particular whole number, we call that set a finite set.

Whenever a set is so large that its cardinal number is not found among the whole numbers, we call that set an infinite set.

2012 Pearson Education, Inc.

Example: Infinite Set

The odd counting numbers are an infinite set.

Word description

The set of all odd counting numbers

Listing method

{1, 3, 5, 7, 9, …}

Set-builder notation

{x|x is an odd counting number}

2012 Pearson Education, Inc.

Equality of Sets

Set A is equal to set B provided the following two conditions are met:

1. Every element of A is an element of B, and

2. Every element of B is an element of A.

2012 Pearson Education, Inc.

Example: Equality of Sets

State whether the sets in each pair are equal.

a) {a, b, c, d} and {a, c, d, b}

b) {2, 4, 6} and {x|x is an even number}

Solution

a) Yes, order of elements does not matter

b) No, {2, 4, 6} does not represent all the even numbers.

2012 Pearson Education, Inc.

Chapter 2: The Basic Concepts of Set Theory

- 2.1 Symbols and Terminology
- 2.2 Venn Diagrams and Subsets
- 2.3 Set Operations and Cartesian Products
- 2.4 Surveys and Cardinal Numbers

2012 Pearson Education, Inc.

Venn Diagrams and Subsets

- Venn Diagrams
- Complement of a Set
- Subsets of a Set
- Proper Subsets
- Counting Subsets

2012 Pearson Education, Inc.

Venn Diagrams

In set theory, the universe of discourse is called the universal set, typically designated with the letter U.

Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these diagrams, the universal set is represented by a rectangle and other sets of interest within the universal set are depicted as circular regions.

2012 Pearson Education, Inc.

U

Venn DiagramsThe rectangle represents the universal set, U, while the portion bounded by the circle represents set A.

2012 Pearson Education, Inc.

U

Complement of a SetThe colored region inside U and outside the circle is labeled A'(read “Aprime”). This set, called the complement of A, contains all elements that are contained in U but not in A.

2012 Pearson Education, Inc.

Complement of a Set

For any set A within the universal set U, the complement of A, written A',is the set of all elements of U that are not elements of A. That is

2012 Pearson Education, Inc.

A

U

Subsets of a SetSet A is a subset of set B if every element of A is also an element of B. In symbols this is written

2012 Pearson Education, Inc.

Example: Subsets

Fill in the blank with to make a true statement.

a) {a, b, c} ___ { a, c, d}

b) {1, 2, 3, 4} ___ {1, 2, 3, 4}

Solution

a) {a, b, c} ___ { a, c, d}

b) {1, 2, 3, 4} ___ {1, 2, 3, 4}

2012 Pearson Education, Inc.

Set Equality (Alternative Definition)

Suppose that A and B are sets. Then A = B if

2012 Pearson Education, Inc.

Proper Subset of a Set

Set A is a proper subset of set B if

In symbols, this is written

2012 Pearson Education, Inc.

Example: Proper Subsets

Decide whether or both could be placed in each blank to make a true statement.

a) {a, b, c} ___ { a ,b, c, d}

b) {1, 2, 3, 4} ___ {1, 2, 3, 4}

Solution

a) both

b)

2012 Pearson Education, Inc.

Counting Subsets

One method of counting subsets involves using a tree diagram. The figure below shows the use of a tree diagram to find the subsets of {a, b}.

Yes

No

{a, b}

{a}

{b}

Yes

No

Yes

No

2012 Pearson Education, Inc.

Number of Subsets

The number of subsets of a set with n elements is 2n.

The number of proper subsets of a set with n elements is 2n – 1.

2012 Pearson Education, Inc.

Example: Number of Subsets

Find the number of subsets and the number of proper subsets of the set {m, a, t, h, y}.

Solution

Since there are 5 elements, the number of subsets is 25 = 32.

The number of proper subsets is 32 – 1 = 31.

2012 Pearson Education, Inc.

Chapter 2: The Basic Concepts of Set Theory

- 2.1 Symbols and Terminology
- 2.2 Venn Diagrams and Subsets
- 2.3 Set Operations and Cartesian Products
- 2.4 Surveys and Cardinal Numbers

2012 Pearson Education, Inc.

Set Operations and Cartesian Products

- Intersection of Sets
- Union of Sets
- Difference of Sets
- Ordered Pairs
- Cartesian Product of Sets
- Venn Diagrams
- De Morgan’s Laws

2012 Pearson Education, Inc.

Intersection of Sets

The intersection of sets A and B, written

is the set of elements common to both A and B, or

2012 Pearson Education, Inc.

Example: Intersection of Sets

Find each intersection.

a)

b)

Solution

a)

b)

2012 Pearson Education, Inc.

Union of Sets

The union of sets A and B, written is the set of elements belonging to either of the sets, or

2012 Pearson Education, Inc.

Difference of Sets

The difference of sets A and B, written A – B,

is the set of elements belonging to set A and not to set B, or

2012 Pearson Education, Inc.

Example: Difference of Sets

Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, h}, B = {c, e, g}, and C = {a, c, d, g, e}.

Find each set.

a)

b)

Solution

a) {a, b, h}

b)

2012 Pearson Education, Inc.

Ordered Pairs

In the ordered pair (a, b), a is called the first component and b is called the second component. In general

Two ordered pairs are equal provided that their first components are equal and their second components are equal.

2012 Pearson Education, Inc.

Cartesian Product of Sets

The Cartesian product of sets A and B, written, is

2012 Pearson Education, Inc.

Example: Finding Cartesian Products

Let A = {a, b}, B = {1, 2, 3}

Find each set.

a)

b)

Solution

a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),

(3, 1), (3, 2), (3, 3)}

2012 Pearson Education, Inc.

Example: Finding Cardinal Numbers of Cartesian Products

If n(A) = 12and n(B) = 7, then find

Solution

2012 Pearson Education, Inc.

B

U

Example: Shading Venn Diagrams to Represent SetsDraw a Venn Diagram to represent the set

Solution

2012 Pearson Education, Inc.

A

C

U

Example: Shading Venn Diagrams to Represent SetsDraw a Venn Diagram to represent the set

Solution

2012 Pearson Education, Inc.

- 2.1 Symbols and Terminology
- 2.2 Venn Diagrams and Subsets
- 2.3 Set Operations and Cartesian Products
- 2.4 Surveys and Cardinal Numbers

2012 Pearson Education, Inc.

Surveys

Problems involving sets of people (or other objects) sometimes require analyzing known information about certain subsets to obtain cardinal numbers of other subsets. The “known information” is often obtained by administering a survey.

2012 Pearson Education, Inc.

Example: Analyzing a Survey

Suppose that a group of 140 people were questioned

about particular sports that they watch regularly and the

following information was produced.

93 like football 40 like football and baseball

70 like baseball 25 like baseball and hockey

40 like hockey 28 like football and hockey

20 like all three

a) How many people like only football?

b) How many people don’t like any of the sports?

2012 Pearson Education, Inc.

F

20

H

Example: Analyzing a SurveyConstruct a Venn diagram. Let F = football, B = baseball, and H = hockey.

Start with like all 3

2012 Pearson Education, Inc.

F

20

Subtract to get

20

8

5

H

Example: Analyzing a SurveyConstruct a Venn diagram. Let F = football, B = baseball, and H = hockey.

2012 Pearson Education, Inc.

F

20

25

Subtract to get

45

20

8

5

7

H

Example: Analyzing a SurveyConstruct a Venn diagram. Let F = football, B = baseball, and H = hockey.

2012 Pearson Education, Inc.

F

20

25

45

20

8

5

7

10

H

Example: Analyzing a SurveyConstruct a Venn diagram. Let F = football, B = baseball, and H = hockey.

Subtract total shown from 140 to get

2012 Pearson Education, Inc.

Analyzing a Survey

- Solution
- (from the Venn diagram)
- a) 45 like only football
- 10 do not like any sports

2012 Pearson Education, Inc.

Example: Analyzing Data in a Table

On a given day, breakfast patrons were categorized according to age and preferred beverage. The results are summarized on the next slide. There will be questions to follow.

2012 Pearson Education, Inc.

Example: Analyzing Data in a Table

2012 Pearson Education, Inc.

Example: Analyzing Data in a Table

Using the letters in the table, find the number of people in each of the following sets.

a) b)

2012 Pearson Education, Inc.

Example: Analyzing Data in a Table

= 15.

- in both YandC

b) not in O (so Y+ M) + those not already counted that are in T

= 55 + 77 + 24 = 156.

2012 Pearson Education, Inc.

Download Presentation

Connecting to Server..