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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

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Symbols and Terminology

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

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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.

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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}

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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

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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}

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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}

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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.

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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)

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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.

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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}

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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.

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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

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Venn Diagrams and Subsets

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

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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.

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U

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

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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.

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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

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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

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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}

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Set Equality (Alternative Definition)

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

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Proper Subset of a Set

Set A is a proper subset of set B if

In symbols, this is written

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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)

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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

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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.

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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.

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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

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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

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Intersection of Sets

The intersection of sets A and B, written

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

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Example: Intersection of Sets

Find each intersection.

a)

b)

Solution

a)

b)

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Union of Sets

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

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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

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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)

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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.

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Cartesian Product of Sets

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

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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)}

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Example: Finding Cardinal Numbers of Cartesian Products

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

Solution

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B

U

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

Solution

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A

C

U

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

Solution

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

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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.

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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?

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F

20

H

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

Start with like all 3

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F

20

Subtract to get

20

8

5

H

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

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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.

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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

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Analyzing a Survey

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

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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.

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Example: Analyzing Data in a Table

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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)

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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.

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