BMM10233 Chapter 8 Set Theory

8/11/2019 BMM10233 Chapter 8 Set Theory

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Chapter 8 | Set Theory | BMM10233 | 137 of 204

collection of well-defined objects is called a set. The objects that constitute the set are known as the elements of the set.

Notation: If x is a member of a set S, it is denoted by x Sx is an element of S. If x is not an element of S it denoted as x

S

Note: The elements of a given set are always contained within brackets.

Example: {1, 2, 3, 4, 5} or {1 2 3 4 5}

Types of sets

1. Null set: set containing no element is known as Null set & it is denoted by or { }.

Remember: {0} or {} is not a null set as they contain the element 0 and respectively.

2. Infinite set: set with infinite number of elements is called an infinite set.

Example: The set of integers is an infinite set.

3. Finite set: set with a finite number of elements is called a finite set.

Example: {1, 2, 3, 4, 5} is a finite set.

4. Disjoint sets: Two sets with no common elements are known as disjoint sets.

Example: = {1, 2, 3, 4, 5} and = {, , 8, 9, 10} are disjoint sets.

5. Equal sets: Two sets with all elements common to both sets are known as equal sets.

Example: = {L, E, , P} and = {P, E, , L} are equal sets as both sets have the same elements.

. Equivalent sets:Two or more sets are equivalent sets if they contain the same number of elements. This does mean that the

elements have to be identical. Only the number of elements is similar.

Example: = {L, E, , P} and = {R, E, , L} are equivalent sets.

. Universal set: The set which contains all the sets under consideration as subsets without duplication of elements is a

universal set.

Notation: The universal set is denoted by U.

8. Subset: set is the subset of a set if every element in the former is contained in the latter.

Notation: If = {1, 2, 3, 4} and = {1, 2, 3, 4, 5, }, then is a subset of as every element of is contained in .

Note:

1 Clearly, the full set and the set itself are subsets of all sets. subset is a proper subset of another set, if there exists at least

one element in the latter that is not contained in the former. Hence excepting the set itself all other subsets are

proper sub sets.

Example: = {L, E, , P}and = {P, E, , R, L} then is a proper subset of because contains at least one element 'R'

which is not contained in .

2 The number of subsets of a set containing 'N' elements is 2N.

Set Theory


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9. Power set: The set of all subset of a given set is known as the power set and is denoted by P .

Example: If = {1, 2, 3}, then the power set of is given by P = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3,}, {}}

10. Complement of a set: The complement of a given set is a set that contains all the elements of the universal set that does not

belong to the given set.

Example: If = {all positive integers} then the complement set of denoted as . = {ll negative integers and zero} if the universal set is the set of integers.

Cardinal Number of a Finite Set

The number of elements contained in a finite set is known as the cardinal number of the set. Cardinal number is not defined

for infinite sets.

Notation: If = {1,2,3} then the cardinal number of set n = 3

Union and Intersection of Sets

1. The union of two sets and denoted by is the set containing all the elements contained in either of the sets withoutrepetition. Note that any element of the set belongs to set or set .

Example: = {1, 2, 3} and = {3, 4, 5}, then = {1, 2, 3, 4, 5}

2. The intersection of two sets and denoted by is the set containing all the elements common to both the sets. Note

that any element of the set belongs to set as well as set .

Example: = {1, 2, 3} and = {3, 4, 5} then = {3}

Difference of Two Sets

The difference of two sets and is the set of all elements that are present in excluding the common elements of and .

Notation: = {1, 2, 3} and = {3, 4, 5}, then the difference of and is - = {1, 2}, - = {4, 5}.

VENN DIAGRAMS

Venn diagram is a pictorial representation of a set or a group of sets and the operations carried out on them. The

representations are standardized with the universal set being represented as a rectangle and all the other sets as circles.

Representation by Venn Diagrams

The rectangular box represents the universal set while the circles represent set and set .

U

A B

U

A B

1. 2.


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U

A B

U

A B

3. 4.

U

A B

5. (A B) . Proper subsets

U

A B

. Disjoint sets

Some Important Relations in Sets

1. n = n + n - n

2. n C = n + n + nC -n - n C -n C + n C


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SOLVED EXAMPLES

Example: 1.

The combined membership of the Lions Club and the Rotary Club in Calcutta is 480. If the Lions Club has 280 members, and

200 members are common to both clubs, then what is the membership of the Rotary Club?

Sol:

8 0 20 0 20 0

L R

The problem can be represented by a Venn diagram.

Number of Common members = 200

Members who belong to Lion Club only = 80.

Therefore, members who belong to Rotary Club only = 480-200 + 80 = 200.

Therefore, Membership of Rotary Club = common members + sole members = 200 + 200 = 400.

Example: 2.

readership survey conducted amongst the general public in angalore revealed that while 5% read the Deccan Herald,

35% read the Hindu. What percentage of the people surveyed, read both the Deccan Herald and The Hindu? ssume that

everyone in angalore reads one of these two newspapers.

Sol:

35-x x 75-x

DH H

From the data given it is observed that the people reading both the newspapers

= 5 + 35 100 = 10%

This can be represented in the form of a Venn diagram

nDH TH = nDH + nTH nDH TH

100% = 5% + 35% - x

x = 5 + 35 100 = 10%.

Example: 3.

If U = {1, 2, 3, 4, 5, , , 8, 9, 10, 11, 12}, = {1, 2, 3, 4, 5, }, = {1, 3, 5, , 9, 11} and C = {2, 4, , 8, 10, 12}

Find the following.

1. 2. C 3. C 4. C

5. . .


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Sol:

1. = {1, 2 ,3, 4, 5, , , 9, 11}

2. C = {2, 4, }

3. = {1, 2, 3, 4, 5, , , 9, 11}

C = {1, 2, 3, 4, 5, , , 8, 9, 10, 11, 12}

4. = {1, 3, 5}

C = { } =

5. = {, 8, 9, 10, 11, 12}

= {1, 3, 5, , 8, 9, 10, 11, 12}

. = {, 8, 9, 10, 11, 12 = {2, 4, , 8, 10, 12}

= {2, 4, , , 8, 9, 10, 11, 12}

. = {2, 4, }

Example: 4.

Company xyz has 0 blue collared workers in its factory and 30 white collared workers at its office. 20 members of the staff

work in both the factory as well as the office. How many people does company xyz employ?

Sol.:

40 20 10

BC WC

Workers in Factory = 0 = .

Workers in Office = 30 = .

Workers in both places = 20 = .

n = n + n n = 0 + 30 20 = 0

Therefore, company xyz has 0 employees.

Example: 5.

survey conducted among 0 school students revealed the following data: 40 drank milk, 30 drank coffee, 24 drank tea, 20

drank milk and coffee and 10 drank coffee and tea, 5 drank milk and tea. How many students drank all three? Every student

drank at least one of the three.

Sol.

10

16

91

14

19

M

C

T


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If M = Milk drinking students, C = Coffee drinking students, T = Tea drinking students, then,

nM nC nT = 0

= nM + nC + nT nM C nM T nC T + nM CT.

0 = 40 + 30 + 24 20 5 10 + nM C T.

nM C T = 1, the number of student drinking all three = 1.

Note: Number of students who drank1 Milk only = 1, 2 Coffee only = 1 3 Tea only = 10.

Example: 6.

In an athletic team, 21 are in the basketball team, 2 are in the hockey team and 29 in the football team, if 14 play both hockey

and basket ball, 12 play both football and basketball, 15 play both hockey and football and 8 play all three, find 1 the total

number of players 2 the number of players who play only football?

Sol :

5

10

6

3

87

4

F

B

H

n = Number of players who play basketball = 21.

nH = number of players in the hockey team = 2.

nF = number of players in the football team = 29.

n H = Number of players who are in both basketball and hockey = 14.

nF = 12, nH F = 15, nF H = 8.

1. Total number of players = n HF

= n + nH + nF n H n F nH F + n HF

= 21 + 2 + 29 14 12 15 + 8 = 43.2. No. of players who play only football = nF n F nF H + nF H =29 12 15 + 8 = 10

Example: 7.

TV survey gives the following data for TV viewing: 0% see program , 50% see program , 50% see program C, 30% see

and , 20% see and C, 30% see and C and 10% do not view any program.

1 What percentage of viewers view all three program?

2 What percentage of viewers view only program ?

Sol:

10

10

20

1010

10

20

C

A

B

Let the total number of viewers be 100

n = number who view program = 0


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n = number who view program = 50

nC = number who view program C = 50

n = 30, n C = 30, n C = 20

1 Percentage of viewers who view all three programs = n C = total no. of viewers no. not viewing {n

+ n + nC C n C} = 100 10 {0 + 50 +5 0 30 20 30} = 10

2 Percentage of viewers who view only = 10.

Example: 8.

In a hostel 45 members drink tea, 24 drink coffee, 18 drink milk, 15 drink both coffee and tea , 12 drink both milk and tea . None

of them drink both coffee and milk.

Find the total number of members.

18

6

15

9

012

00

M

C

T

Sol:

Total number of members = 0

Note : Since none of the members drink both coffee and milk, there cannot be any member which drinks all the three.

Total number of members = 45 + 24 + 18 15 12 = 0 by formula

Example: 9.

If U = {set of all numbers 0 to 100 both inclusive}

= {set of positive numbers upto 100 and divisible by 4}

= {set of positive two digit numbers} then find

1 2 3 4 U 5. U

Sol:

U

= {set of positive number upto 100 and divisible by 4}

{4, 8, 12, 1, 20, 24, 28, 32, 3, 40, 44, 48, 52, 5, 0, 4, 8, 2, , 80, 84, 88, 92, 9, 100}


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so set a contains 25 number.

= {set of positive 2 digit number}

{10, 11, 12, ..., 99} 90 numbers.

1 = {4, 8, 10, 11, 12, 13, ..., 99, 100}

UA B

2 = {12, 1, 20, ..., 88, 92, 9}

UA B

3 =

UA B

= {0, 1, 2, ..., 100} {4, 8, 10, 11, ..., 99, 100}

4

UA B

= {4, 8, 12, 1, ..., 9, 100} {0, 1, 2, ..., 100}


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= {4, 8, 12, 1, ..., 9, 100}

5 =

UA B

shaded area is the answer

= {4, 8, 12, 1, ..., 92, 9, 100} {10, 11, ..., 99} {0, 1, 2, ..., 100}

= {12, 1, ..., 9}

= {4, 8, 12, ..., 9} {12, 1, ..., 92, 9}

Example: 10.

Find the result of the following:

1 2 3 4 5

Sol:

1

( }

UAA U

2

{}

UAA U

3

{}

UAA U


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4

UAA U

5

UAA U

Example: 11.

In a sample of 400 people, it was found that 200 own an HMT watch, 0 own a Titan and 50 own llwyn Quartz. 30 own both

a Titan and HMT, 20 own an HMT and llwyn and 8 own Titan and llwyn, 4 own all three. Find out:

1 How many people from the sample do not own a watch? 2 How many own an HMT alone?

3 How many own a Titan alone? 4 How many own an lwyn alone?

Sol:

Titan

36

AllwynQzalrtz

26HMT154

4 416

26

Total 154+ 2 + 3 + 4 + 4 + 1 + 2 = 2

1 Number of people not owning a watch 400 - 2 = 134. 2 154 own HMT alone.

3 3 own Titan alone. 4 2 own allwyn alone.

Example: 12.

In the foreign department of the University of Oklahoma, 40 students study Spanish, 30 study rabic and 20 study Chinese.

If 20 students study Spanish only, 14 study rabic only, 10 study Spanish and rabic and 4 study all three, then find number

of students who

1 Study Chinese only?

2 Study Chinese and Spanish?

3 Study Chinese and rabic?

4 Study Spanish and rabic but not Chinese?

5 Study Chinese and Spanish but not rabic?


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Sol:

Arabic 1410

Spanish 20 4 2

6 8Chinese

1 8 students study Chinese only.

2 10 students study Chinese and Spanish

3 students study Chinese and rabic

4 10 students study Spanish and rabic but not Chinese

5 10 students study Chinese and Spanish but not rabic

Example: 13.

salesman visits 24 housewives and finds that 15 use , 98 use only , 14 use and C but not , 39 use & C, 48 use only

and 22 use all three - , , and C being brands of detergent.

a Which detergent is most popular according to his inquiry?

b How many use only C? c What fraction uses at least two products?

Give

x + y = 39

x = 22

48

BZA98

X Y14

C

1

Sol:

98+ 14 + x + z = 15

z = 23

89 + z + 48 + x + y + 14 + c = 24.

98 + 23 + 48 + 22 +1 + 14 + c = 24.

222 + c = 24. c = 24 222. c = 52

48

B23A

9822

17

14

52

C


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a Compare

98 + 23 + 14 + 22, = 15,

48 + 23 +22 + 1 = 110,

14 + 22 + 1 + 52 + = 105,

Detergent is popular.

b 52 use only c

c76

0.28274

=

Example: 14.

In an examination, question 8 was attempted by candidates, question 9 by 4

and question 10 by 40. 28 candidates attempted 8 and 9, 8 attempted both 9 and 10, 2 attempted both 8 and 10 and 2

attempted all three. How many attempted

1 only question 8? 2 only question 9?

Que.

ZYQue. X

B CA

Que. 10

Given.

x + y + + =

y + z + + c = 4.

+ + C + D = 40.

Y = 28

C = 8 = 2

= 2

Sol:

1 x + 28 + 2 + 2 =

x = 11

2 Y + Z + + C = 4

28 + Z + 2 + 8 = 40.

Y = 8

Example: 15.

survey in a gym showed the following: 30% work on their legs, 0% work on their biceps, 0% work on their chests, 20%on legs and biceps, 30% on biceps and chest, 10% on legs and chest, 5% work on all the three. Find how many

1 work on only legs and biceps?

2 work on exactly two body parts?


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Sol:

Biceps 15%15%Legs

5%5%

25%5%

25%Chest

1 15% 2 45%

Example: 16.

survey of Hollywood showed the following: 0% drove Lamborghinis, 50% drove Ferraris, 50% drove Rolls, 30% drove

Lamborginis and Ferraris, 20% Ferraris and Rolls, 30% Lamborginis and Rolls, 10% all three, while the rest drove only

Mercedes. What percent drove

1 Lamborginis and Ferraris but not Rolls? 2 Exactly two of the three? 3 Only rolls?

Sol:

F

10%20%L

10% 10%

10%20%

1 20% 2 50% 3 10%

Exaple: 17.

survey shows that 3% of the Indians like cheese whereas 5% like apples. If x% of the Indians like cheese and apples, find

the value of x.

75%63%

X%Apples

Chees

Sol:

3% + 5% x% = 100%2 + 5 100 = x

x = 38


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Example: 18.

If , and C are three sets and U is the universal set such that n U = 00. n = 200, n = 300 and n = 100. Find

n.

Sol:

U AB

n = 00

n = 200

n = 300

n = 100

n = 00

Find nn = n n

= 00 400

= 300

Example: 19.

college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to a total of 58 men and only

three men got medals in all the three sports, how many received medals in exactly two of the three sports?

B

ZYF

x BC

A


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Given

X + Y + + = 38

Y + Z + + C = 15

+ + C + D = 20

= 3.

+ Y + Z + + + C + D = 58

+ Y + C = ?Sol:

+ Y = 35

Y + C = 12

+ C = 1

2 + Y + C = 35 + 12 + 1 = 4

+ Y + C = 32

Example: 20.

class has 15 students. The following is the description showing the number of students studying one or more of the

following subjects in the class.

Mathematics 100; Physics 0; Chemistry 4; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and

Chemistry 23; Mathematic, Physics and Chemistry 18. Findi How many students are enrolled in Mathematic alone, Physics alone and Chemistry alone?

ii The number of students who have not enrolled in any of these three subjects?

Sol:

351260

185

10

Mathematics

Physics

Chemistry

i Mathematics alone = 0

Physics alone = 35

Chemistry = 13

ii 15 0 + 12 + 35 +18 + 10 + 5 + 13 = 22


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1. If U = {2, 3, 4, 5, , , 8, 9, 10, 11}, = {2, 4, }, = {3, 5, , 9, 11} and C = {, 8, 9, 10, 11}, compute : U C. ccess

Code - 01308001

1 {} 2 {9} 3 {} 4 {5} 5 {4}

2. If U = {a. b, c, d, e, f}, = {a, b, c}, find U '? ccess Code - 01308002

1 U 2 3 4 oth U and 5 None of these

3. If U = {2, 3, 4, 5, , , 8, 9, 10, 11}, = {3, 5, , 9, 11} and = {, 8, 9, 10, 11}, compute '. ccess Code - 0130003

1 {2, 3, 5, , 9, 11, 12} 2 {2, 4, , 8, 10, 11, 12}3 {2, 4, , , 8, 9, 10, 11} 4 {2, 3, , 8, 9, 11, 12} 5 None of these.

4. If X and Y are two sets such that X Y has 18 elements. X has 8 elements, and Y has 15 elements, how many elements does

X Y have? ccess Code - 01308004

1 5 2 3 9 4 11 5 10

5. If and are two sets such that has 40 elements, has 0 elements , and has 10 elements, how many elementsdoes have? ccess Code - 01308005

1 40 2 30 3 45 4 50 5 30

. If S and T are two sets such that S has 21 elements, T has 32 elements, and ST has 11 elements, how many elements, does

S T have? ccess Code - 0130800

1 52 2 32 3 42 4 45 5 50

. In a group of 1000 people, there are 50 people who can speak Hindi and 400 who can speak English. How many can speak

Hindi only? ccess Code - 0130800

1 00 2 50 3 50 4 800 5 00

8. In a class of 50 students, 35 opted for Mathematics and 3 opted for iology. How many have opted for both Mathematics

and biology? ssume that each student has to opt for at least one of the subjects. ccess Code - 01308008

1 15 2 1 3 25 4 13 5 22

9. In a survey, it was found that 5% of the people watched news on TV, 40% read in newspaper, 25% read newspaper and

watched TV. What percentage of people neither watched TV nor read newspaper? ccess Code - 01308009

1 0% 2 5% 3 10% 4 30 % 5 20%

10. In a town with a population of 5000, 3200 people are egg-eaters, 2500 meat eaters and 1500 eat both egg and meat. How

many are pure vegetarians? ccess Code - 0130801

1 00 2 800 3 900 4 850 5 1000

11. In a group of 15 woman, have nose studs, 8 have ear rings and 3 have neither. How many of these have both nose studs

and ear rings? ccess Code - 01308011

1 0 2 2 3 4 4 5 3

Practice Exercise - 1


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12. IF = {a, d}, = {b, c, e} and C = {b, c, f}, then C = ccess Code - 01308012

1 2 C 3 C 4 Data insufficient 5 None of these

13. If = {a, d}, = {b, c, e} and C = {b, c, f}, then C = ccess Code - 01308013

1 C 2 3 C 4 C 5 None of these

14. There are 19 hockey players in a club. On a particular day 14 were wearing the prescribed hockey shirts, while 11 were

wearing the prescribed hockey pants. None of them was without hockey pant or hockey shirt. How many of them were in

complete hockey uniform? ccess Code - 01308014

1 8 2 3 9 4 5 10

15. ll the students of a batch opted Psychology, usiness or both. 3% of the students opted Psychology and 2% opted

usiness. If there are 220 students, how many of them opted for both Psychology and usiness? ccess Code - 01308015

1 0 2 100 3 80 4 35 5

SCORE SHEET

Use HB pencil only. Abide by the time-limit

1

2

3

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

4

5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

8

9

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

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1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

13

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1 2 3 4 5

1 2 3 4 5

1 2 3 4 5


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1. In a class of 100 students, the number of students passed in English only is 4, in Maths only is 4, in Commerce only is 58.

The number who passed in English and Maths is 1, Maths and Commerce is 24 and English and Commerce is 2, and the

number who passed in all subjects is . Find the number of students who failed in all the subjects? ccess Code -

0130801

1 2 8 3 10 4 5 9

Directions for (Que. 2 - 8): Refer to the data below and answer the questions that follow.

In an examination 43% passed in Maths, 48% passed in Physics and 52% passed in Chemistry.

Only 8% students passed in all the three. 14% passed in Maths and Physics and 21% passed in Maths and Chemistry and

20% passed in Physics and Chemistry. Number of students who took the exam is 200.

Let set P, C and M denote the students who passed in Physics, Chemistry and Maths respectively.

2. How many students passed in Maths only? ccess Code - 0130801

1 1 2 32 3 48 4 80 5 50

3. Find the ratio of students passing in Maths only to the students passing in Chemistry only ccess Code - 01308018

1 1 : 3 2 29 : 32 3 1 : 19 4 31 : 49 5 None of these

4. What is the ratio of the number of students passing in Physics only to the students passing in either Physics or Chemistry

or both? ccess Code - 01308019

1 33/4 2 22/80 3 49/32 4 53/3 5 None of these

5. student is declared pass in the exam only if he/she clears any two subjects. The number of student who were declared

passed in this exam is? ccess Code - 01308020

1 33 2 3 39 4 8 5 80

. If = {1,{2, 3},5}, which of the following statements is incorrect ccess Code - 01308021

1 {2,3} A 2 {{2,3}} A 3{3} A 4 {5} A 5 None the these

. Which one of the following is a correct statement? ccess Code - 01308022

1 every subset of an infinite set is infinite

2 every set has a proper subset

3 {a, b, c, 1, 2, 3, a, b, c, 1, 2, 3, ...} is an infinite set

4 every subset of a finite set is finite

5 None of these

8. If = {1, 2, 3, .... 9}, = {2, 4, , , 8} and C = {3, 4, 5, 8, 9, 10}, then - C is: ccess Code - 013080231 {1, 3, 4, 5, 8, 9, 10} 2 {1, 2, 3, 4, 5, , , 8, 9}3 {2, 4, , , 8} 4 {1, 3, 4, 5, 8, 9} 5 {1, 2, 3, , 8, 9}

Practice Exercise - 2


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9. In a survey of brand preference for toothpaste, 82% of the population number of people covered for the survey liked at

least one of the brands : I, II and III. 40% of those asked liked brands I, 25% liked brand II and 35% liked brand III. If 5% of

those asked, showed liking for all the three brands, then what percentage of those asked liked more than one of the three

brands? ccess Code - 01308024

1 13 2 10 3 8 4 5 5 15

10. How many ml of water must be added to 48 ml of alcohol to make a solution that contains 25% alcohol? ccess Code -

01308025

1 48 2 4 3 144 4 192 5 90

SCORE SHEET

Use HB pencil only. Abide by the time-limit

1

2

1 2 3 4 5

1 2 3 4 5

3

4

1 2 3 4 5

1 2 3 4 5

5

1 2 3 4 5

1 2 3 4 5

8

1 2 3 4 5

1 2 3 4 5

9

10

1 2 3 4 5

1 2 3 4 5


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3. 3

4. 2

5. 4

. 1

. 4

8. 1

9. 1

10. 3

4. 1

5. 5

. 3

. 1

8. 5

9. 5

10. 2

11. 3

12. 3

13. 1

14. 2

15. 5

1. 5

2. 2

1. 1

2. 3

3. 3

Answer Key

Practice Exercise -1

Practice Exercise -2

Documents

BMM10233 Chapter 8 Set Theory