topic 3 sets

8/3/2019 Topic 3 Sets

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INTRODUCTIONThe modern mathematical theory of sets is one of the most remarkable creations

of in the field of mathematics. This is because of the unusual boldness of the ideas

found in its theory. But above this, the theory has assumed tremendous

importance for almost the whole of mathematrics. In this topic, we will learn a

few key ideas from set theory. Set concepts and notation not only help us talk

about certain mathematical ideas with greater clarity and precision, but are

indispensable to a clear understanding of probability.

SET

3.1.1 Sets

3.1

A set is a well-defined collection of objects, called elements or members.

TTooppiicc

33Sets

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Define sets and subsets;2. Describe the set characterictics and notation;3. Visualise the set operations using Venn diagrams; and4. Analyse statistical survey using the method of counting the number o

elements in a set.


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The main property of a set in mathematics is that it is well-defined. This means

that given any object, it must be clear whether that object is a member of the set.

Example 3.1Which of the following satisfy the requirements of a set?

(a) all the current United States senators from Massachusetts(b) all the prime divisors of 1987(c) all the tall people in Canada(d) all the prime numbers between 8 and 10(e) all the funny comics in the daily newspaper(f) all the students taking mathematics at Open University Malaysia at thepresent moment(g) all the good writers in Malaysia(h) all the beautiful girls in the classDescriptions (a), (b), (d) and (f) are well-defined and therefore define sets.

Descriptions (c), (e), (g) and (h) are not well defined and therefore do not define

sets.

It is customary to use capital letters to designate sets and the symbol to denotemembership in a set. The members of a set are listed in braces and to separatethese members with commas. Thus we write a A to mean that object a is amember of setA and to mean that the object a is not a member of setB.a B

Example 3.2

If , then{1, 2,3,4}A 2 A and 5 A .

3.1.2 Subsets

When considering two setsA andB, it may happen that every element ofA is an

element ofB

LetA andB be sets. A is a subset ofB (orA is contained in B) if every elements of

A is an element ofB, and we denote this by writing . In order to visualise

the concept of set , we can make use the Venn diagram as shown in Figure 3.1.

A B


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Figure 3. 1

Example 3.3

If { ,{ }} A a b , then its subsets would be

{ , { {{ and { }}a b }a }}b

The symbol { or} represents the empty or null set

The empty set is subset of every set.

Notice that there is a set { contained inA and this set is an element ofA

and not a subset ofA. In order to be a subset of it must written as {{ .

}b

}}b

Figure 3.2

A setA is said to be a proper subset ofB denoted by , ifA is a subset ofB

but . In other word, means that all elements ofA are also inB, but

B contains at least one element that is not inA. (Figure 3.2)

A BA B A B

Example 3.4

If { , } A a b , then its subsets would be

{ { and }a }b


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Example 3.5

Assume and{{1, 2},{1, 2,3},1,2,3}X {{1, 2},1,2,3,{2,3}}Y . Which of the

following statements are correct.

(a) {2} X Y (b) {2} X Y (c) {1, 2} ( )X Y (d) {1, 2,3} ( )X Y (e) {{1, 2}} ( )X Y (f) 2 ( )X Y Solution

Before answering these questions, the resulting set from the operation X Y must first be obtained.

{{1, 2},{1, 2,3},1, 2,3} {{1, 2},1, 2,3,{2,3}} {{1, 2},1,2,3}X Y

Statement True/False Explanation

X Y {2}{2} T is a subset ofX Y

{3} X Y F {3} is a subset of X Y and not an

element ofX Y .

{1,2} ( )X Y T {1,2} is an element ofX Y

{1,2,3} ( )X Y T {1, 2,3} is a subset ofX Y

{{1, 2}} ( )X Y F {1,3} is a subset of X Y and not an

element ofX Y

2 ( )X Y F 2 is an element X Y and not a subsetofX Y


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3. 1.3 Defining A Set

Sets can be stated in three ways:

By giving a written description of the set By listing the elements of the set within braces (roster method) By using set builder notation the values of its digits

Example 3.6

Written Description Roster Notation Set Builder Notation

The set of natural

numbers

{1, 2, 3,............} {xlx is a natural number}

The set of integers {...-2,-1, 0,1,2,...} {xlx is an integer}

The set of prime

numbers

{2, 3, 4,...} {xlx is a prime number}

The set of irrational

numbers 2 3, 5, 7, 11, 19, {xlx is an irrational number}

The set of decimalnumbers

{2.2, 1.35, 3.45, 5.47,6.9} {xlx is a decimal number}

In set-builder notation, the vertical bar is used to mean such that and the words

to the right of the bar describe the rule.

Example 3.7

(a) {xlx is a natural number} is read as the set of all x such that x is naturalnumber.

(b) {xlx is a prime number} is read as the set of all x such that x is a primenumber.

Why do we have three types of notation for writing sets?

SELF-CHECK 3.1


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3.1.4 Universal Sets

The universal set is the set of all elements under discussion.

Example 3.8

Find all the subsets of the set { , , }a b c

Form all the subsets with no element Form all the subsets with one element { } , { }, { }a b c

Form all the subsets with two elements { , },{ , }, { , }a b b c a c

Form all the subsets with three elements { , , }a b c

The set in the above example has 3 elements. So this set has 8 subsets.

In general, if the set has n elements, then the number of subsets of the set is .2n

In the above example the set has 3 elements, then its number of subsets is .3

2 8

but the number of its proper subsets is3

2 1 2 1 7n . Why?

1.

Which of the following satisfy the requirements of a set?(a) All retired baseball players with lifetime batting averages of

400 or batter

(b) All even numbers can be divided by 2(c) All the smart students in SBMA 1103 class(d) All the prime numbers between 8 and 10(e) All students taking mathematics courses at Open University

at the present moment

(f) {xlx is a good college course}(g) {xlx is an odd counting number}

EXERCISE 3.1


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2. List all the subsets and indicate which are proper subsets of the givenset.

(a) { , }a b (b) { , }a b c, (c)

3. A set has 31 proper subsets . How many elements are there in the set?4. Is it true is a subset of ?

INTERSECTIONS AND UNIONS

3.2

3.2.1 Intersection of Sets

IfA and B are sets, the intersection ofA and B denoted by A B (read as AintersectionB), is the set of all elements that are common to bothA andB. That is

A B {xl x A and x B }

Figure 3.3

Example 3.9

If and{1,2,3,4}A {2,4,6}B , then {2, 4}A B .


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3.2.2 Union of Sets

IfA andB are sets, the union ofA andB is denoted byA B (readA unionB). It

is the set of all elements that are either in A or in B or in both A and B whichrepresented by Figure 3.4.

A B {xl x A orx B }

Figure 3.4

Example 3.10

If and , then .{1, 2,3,4}A {2,4,6}B {1,2,3,4,6}A B

Disjoint Set

Two sets with no elements in common are said to be disjoint. If setA andB are

disjoint, then A B

Figure 3.5

Example 3.11

If and , then{1, 2,3,4}A {5,6,7}B A B .

3.2.3 Complement of a Set

Let be universal set, then letA be a subset of. The complement ofA, denoted

(read A prime or A complement) is the set of all elementA that are not in

A. This set is symbolised by A .


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Figure 3.6

Example 3.12

Let { , , , , , }a b c d e f , and{ , , }A a c e { , }B b d . Find the following:

(a) A (b) B Solution

(a) { , , }A b d f

(b) { , , , }B a c e f Example 3.13

Let { , , , , , }a b c d e f , and{ , , }A a c e { , , }B b d f . Find the following:

(a) A B (b) A B (c) ( )A B (d) A B (e) A B (f) ( ) A A B Solution

(a) { , , } and { , , }A b d f B a c e { , , } { , , }

{ , , , , , }

A B b d f a c e

a b c d e f


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(b) { , , } { , , }A B a c e b d f

{ , , , , , }

( )

a b c d e f

A B

(c) }( ) { , , } { , ,A B a c e b d f

( ) { , , , , , }A B a b c d e f

(d) { , , }a c e { , , }A B a c e { , , }a c e

(e) { , }{ , , } ,A B b d f a c e

(f) , , , }f ( ) { , , } { , ,A A B b d f a b c d e { , , }b d f B

Notice that the answers to parts a) and c) are identical that is ( ) A B A B

( )

.

Also notice the answers to parts b) and e) are identical that is A B A B .

In general, we have De Morgan Law which states that:

It is possible to form unions, intersections and complements using more than two

sets.

Example 3.14

Let { , , , , , }a b c d e f , { , , }A a c e , { , , }B b d f and C a{ , }d . Find

.( ) A C B

Solution

Since A C is in parentheses, we have to find A C first.

{ , , } { , } { , , , }A C b d f a d a b d f

Then

{ , , }B a c e

For any setsA andB, and (( ) A B A B ) A B A B


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Hence

( ) { , , , } { , ,

{ }

A C B a b d f a c e

a

}

3.2.4 Difference of Two Sets

IfA andB are two sets , the difference ofA andB, denoted by A B (readA minusB) is the set of all elements that are inA and not inB. That is,

A B { xl x A andx B }.

Example 3.15

Let { , , , , , }a b c d e f , and{ , , , }A a b c d { , , }B a b e . Find

(a) A (b) A (c) A B (d) A B

Solution

(a) A is the set of all elements in and not in A , that is { , }e f(b) A is the set of all elements in and not in A , that is { , }e f (c) A B is the set of all elements in A and not in B , that is { , }c d (d) B A is the set of all elements in B and not in A , that is { }e (e) A B is the set of all elements in A and not in B , that is { , }c d Note that

1. The definition of A is a special case of the above definition becauseA A

2. A B A B , because A B is the set of all elements inA and not inB,and this is precisely the definition ofA B


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Example 3.16

Based on the Venn diagram below

state which statements are true.

The shaded region represents the set

(a) A B (b) B A (c) A B (d) A B (e) A B Solution

The correct region is d) i.eA B .

EXERCISE 3. 2

{ , , , , }a b c d e1. Let the sets , { , , }A a c e , B { , , , }b d e f and

C a .{ , , , }b d f

A B

Find the following:

(a)

(b) B C (c) A (d) A ( )B C

( ) ( )

A(e) B B C


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2. Based on Question 1, find the following

(a) A(b) B (c) ( )A B (d) A B (e) ( ) A B C

A ( )B C

{{ , },{ ,

(f)3. Assume , }, , }X x y x y z x y and ,{ , }}Y y z{{ , }, , , ,x y x y z .

Which of the following statements are correct.

(a) Y { }y Xy Y(b) b). { } Z

(c) c). { , ( )x } y X Y , }

(d) d) { , ( ) x y z Y )

X

(e) e) 2 (X Y 2 ( )

(f) f). X Y {1,2

4. Let the sets ,3,4,5}

NUMBER OF ELEMENTS IN A FINITE SET

3.3.1 Number of Elements in a Finite Set

Simple counting techniques usually involve the counting of elements in a given

set. IfA is any set, the number of elements inA is denoted by n(A). Ifn(A) is a

whole number, the set is a finite set.

3.3

, {2,3,5}A , {1,3,4}B and

C a b d .


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Example 3.17

If , then{ , , , , , }A k a n c i l ( ) 4n A . Likewise, if { , , , , , , , }B p e l a n d o k , then

.( ) 8n B

We shall be interested here in counting the number of elements in sets formed by

the operations of union, intersection, and taking complements.

Example 3.18

In Open University Malysia, 1000 students are registered for Algebra course, 500

are registered for Calculus course and 300 are registered for both courses.

(a) What is the total number of registered students?(b)

How many students are taking Algebra course only?

(c) How many students are taking Calculus course only?Solution

Lets represent the information in Venn Diagram as shown below.

Figure 3.7

Let A be the set of students registered for Algebra, and C be the set of student

registered for Calculus.

Step 1

A C has 300 students. Write 300 in the region corresponding to .A CFirst we have to draw a Venn diagram with overlapping region to show the

information.

Step 2

Since ( ) 1000n A and 300 in the intersection ofA and C, the number taking

Algebra courses only is 1000-300 = 700 as shown in Figure 3.7.


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Step 3

The number of students taking Calculus course only is 500 300 = 200.

(a) The number of registered students,( ) 700 300 200 1200n A C .

(b) The number of students taking only the Algebra course is 700.(c) The number of students taking only the Calculus course is 200.Now, we have

( ) ( ) ( ) ( )n A C n A n C n A C

1200 1000 500 300

In general, we can find the number of elements in the union of two sets by

using the formula as stated below.

Assume A and B are any two given sets. There are two possibilities to be

considered

: In this case( )A B ( ) ( ) ( )n A B n A n B

( )A B ( ) ( ) ( ) ( )n A B n A n B n A B : In thiscase

Example 3.19

If ,( ) 66n A ( ) 36n B and n A , find .( ) 12B ( )n A B

Solution

By using the above formula, n A ,( ) ( ) ( ) ( )B n A n B n A B

66 36 12 90

Example 3.20

In a recent survey of 110 college students, the number of students taking Algebra

(A), English(E), and Geography (G) are shown in Figure 3.8.


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Figure 3.8

(a) How many students are taking Algebra or English, but not both?(b) How many students are taking Algebra or English, but not Geography?(c) How many students are taking one or two of these courses, but not all three?(d) How many students are taking at least two of these courses?(e) How many students are taking at least one of these courses?Solution

(a) From the Venn diagram the number of elements inA orE, but not in both31 + 10 + 30 + 15 = 86

(b) The number of students inA andE, but not in G is31 + 8 + 30 = 69

(c) The number in the entire universal set is 110 minus the number taking allthe three courses, 3, or none of these courses, 4. The result is

110 3 4 = 103

(d) The required number of students is in( ) ( ) ( )A E A G E G

Hence, the number is

10 + 8 +15 + 3 = 36


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(e) We can get the required number of the students by taking the number in theuniversal set minus the number taking none of the courses.. Hence the

number is

110 4 = 106

1. IfA and B are sets, the intersection ofA and B denoted by A B (read AintersectionB), is the set of all elements that are common to both A and B.

Symbolically

andA B xl x A x B 2. Let A and B be sets. A is a subset ofB (orA is contained in B) if every

elements ofA is an element ofB, and we denote this by writing A B .

3. The universal set is the set of all elements4. IfA and B are sets, the intersection ofA and B denoted by A B (read A

intersectionB), is the set of all elements that are common to both A and B.

That is

andA B xl x A x B 5. IfA andB are sets, the union ofA andB denoted by A B (readA unionB),

is the set of all elements that are either inA or inB or in bothA andB That is

orA B xl x A x B 6. Let be universal set, then let A be a subset of . The complement ofA,

denoted 'A (read A prime or A complement) is the set of all element

that are not inA. This set is symbolized by A .

7. The Venn diagram help us to visualise the abstract concept of a set.


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Builder notation

Complement of a set

Difference of two sets

Finite set

Intersection of sets

Proper subset

Set

Subset

Union of sets

Universal set

1. Suppose that ,( ) 30n A ( ) 35n B and ( ) 1n A B 4 . Find .( )n A B

2. Suppose that ,( ) 28n A ( ) 8n B and . Find .( )n A B 31

5 4

( )n A B

3. Suppose that , and( ) 30n A ( ) 4n A B ( ) 1n A B . Find .( )n B

4. Suppose that , and( ) 30n A ( ) 4n A B 5 ( ) 15n B .

Find. (n A B) . What can you say about setA and setB?

1.

Figure 3.9

In a survey 120 investors to see who owned gas company Stock (G),

Transportation stock (T) or petroleum bond (P), the numbers shown in the

diagram were found.


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(a) How many investors owned gas company or transportation stock, but notboth?

(b) How many investors owned gas company or transportation stock, but notpetroleum bond?

(c) How many had one or two of these types of investments, but not all three?(d) How many had at least two of these types of investments?(e) How many had none of these types of investments?2.

Figure 3.10

A number of housewives were interviewed to find out who buys product A, B

and C regularly. The result is shown in the Venn diagram above.

(a) How many buy product A?(b) How many buy product A but not B?(c) How many buy product B or C, but not A?(d) How many do not buy product C?(e) How many housewives were interviewed?

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