01. theory of set

7/27/2019 01. Theory of Set

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Theory of Set

Table of Contents

2.1 Set ................................................................................................................................... 2

2.2 Elements of a set ............................................................................................................. 2

2.3 Methods of describing a Set............................................................................................. 2

2.4 Types of Sets ................................................................................................................... 2

2.4.1 Finite Set: ................................................................................................................. 2

2.4.2 Infinite Set: ............................................................................................................... 2

2.4.3 Singleton / Unit Set: .................................................................................................. 2

2.4.4 Empty, Null or Void Set............................................................................................ 2

2.4.5 Equal Set .................................................................................................................. 2

2.4.6 Equivalent Sets: ........................................................................................................ 3

2.4.7 Subsets...................................................................................................................... 3

2.4.8 Proper set .................................................................................................................. 3

2.4.9 Family of Sets ........................................................................................................... 4

2.4.10 Power Set ................................................................................................................ 4

2.4.11 Universal Set........................................................................................................... 4

2.4.12 Disjoint set .............................................................................................................. 4

2.5 Venn Diagrams: .............................................................................................................. 4

2.6 Operation of Sets: ............................................................................................................ 5

2.6.1 Union of Sets: ........................................................................................................... 5

2.6.2 Intersection of Sets:................................................................................................... 5

2.6.3 Difference of Two Sets ............................................................................................. 5

2.6.4 Complement of Set ................................................................................................... 5

2.7 Symmetric Difference ..................................................................................................... 6

2.8 Algebra of Sets: ............................................................................................................... 6

2.9 Number of Elements in a Finite Set ................................................................................. 7

2.10 Cardinality .................................................................................................................... 8

2.11 Cartesian Product: ......................................................................................................... 8

2.12 Common notations ........................................................................................................ 8

2.13 Problems on Set Theory ................................................................................................ 9


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2.1 Set

A set is a collection of well-defined and well-distinguished objects. For example:

The possible outcomes in the toss of a die The integers from 1 to 100 The vowels in English Alphabets.

2.2 Elements of a set

X A

X epsilon A X is a member/element of the Set A X belong to A

2.3 Methods of describing a Set

There are Two Approaches

i. Tabular, Roster or Enumeration Method. Example: A = {a, e, i, o, u}ii. Selector, Property Builder or Rule Method. Example: A = (X : X is a vowel in

English Alphabet}

2.4 Types of Sets

2.4.1 Finite Set: The Element Of A Set Can Be Counted By A Finite Number.

A = {1, 2, 3, 4, 5, 6,}

B = {1, 2, 3, 4..500}C = {X: X is an even positive Integer


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A = B if and only if {X AX B}For example,

Let A = {3, 5, 5, 9}, B= {9, 5, 3}; Then A = B since each element which belong to Abelong to B and each elements of B also belong to A.

Let A = {2,3}, B ={3,2,2,3 and C = {X : X-5X+6=0}; This A=B=C since each elementwhich belong to any one of the sets also belongs to the other two sets.

[Order and Number of element is not important]

2.4.6 Equivalent Sets: if the elements of one set can be put into one to one correspondence

with the elements of another set. Numbers of elements are equal.

For example: A = {1, 2, 3,}, B = {a, b, c}, C = {1, 1, 2, 3} Here, A B; But, A = C.2.4.7 Subsets: if every element of a set A is also an element of a set B then set A is called

subset of set B. Symbolically we write this relationship as A B; and is read as Ais asubset of B or A is contained in B or A is included in B. Sometime this relationship iswritten as B A and is read as B is a superset of A or B Contains A or B includes A.

It may be noted that A B means that every element of A is also an element of B and there

is no restriction on set B other than it includes the set A. Thus set A may be smaller than set

B, when it contains some (not all) the elements of B. Set A may be equal to set B, when it

contains all the elements of B.

For example: A = {1, 2, 3, 4}, B = {1, 2, 3}, C = {1, 2}, D = {1, 2, 3, 4}.So B A, C A, C B, D A

2.4.8 Proper set: Set A is called proper subset of superset B if each and every element of set

A are the elements of the set B and at least one element of superset B is not an element of set

A. Symbolically this is written a A B and is read as A is a proper subset of superset B.

For example A ={1, 2, 3, 5, 9, 12}, B ={1, 2, 2, 3, 5, 9, 12}, C ={1, 2, 2, 3, 3, 5, 9};So C A, A B, B A, and A = B

Remarks:

i. The symbol and or and are inclusion symbols.ii. The larger set is always at the open end of the sign or and or .

iii. If A is a subset of B, then B is called a superset of A.iv. If A is not a subset of B then, there exists at least one element in A which is not a

member of B.

v. A set is always a subset of itself. Ex: A A.vi. The null set is a subset of every set.

vii. If A is a subset of B and B is a subset of C then A is a subset of C. Ex: A B and BC A C.

viii. If A is subset of B and B is subset of A then the sets are said be equal. Ex: AB,BA A = B.

ix.

If A

, then A = .


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2.4.9 Family of Sets: If all the elements of a set are sets themselves then it is called a set ofsets or better term is a family of sets. For example, if A = {a, b} then the set A = {, {a},{b}, {a, b}} is a family of sets whose elements are subsets of the sets A.

2.4.10 Power Set: From a set containing n elements, 2

n

subsets can be formed

.

The setconsisting of these 2nsubsets is called a power set. In Other words, if A be a given set then

the family of sets each of whose number is a subset of the given set A is called the power set

of set A and is denoted as P(S). For Example,

If A = {a}, then its subsets are , {a}.P(S) = [, {a}]

If A = {a, b}, then its subsets are , {a}, {b}, {a, b}P(S) = [, {a}, {b}, {a, b}]

If A = {a, b, c}, then its 23subsets are , {a}, {b}, {c}, {a, b}, {b, c}, {c, a} {a, b, c}P(S) = [, {a}, {b}, {c}, {a, b}, {b, c}, {c, a} {a, b, c}]

2.4.11 Universal Set: In set theory, a universal set is a set which contains all objects,

including itself. The Universal Set will generally be denoted by the symbol U. ForExample: A set of integers may be considered as a universal set for a set of odd or even

integers.

2.4.12 Disjoint set: Two sets are said to be disjoint if they have no element in common. For

example, {1, 2, 3} and {4, 5, 6} are disjoint sets.

2.5 Venn Diagrams:The Venn diagrams are named after English logician and lecturer in Moral Science JohnVenn (1834 - 1923) to present pictorial representation. The universal set, say U is denoted by

a region enclosed by a rectangle and one or more sets say, A, B, C are shown through circles

or closed curves within these rectangles. These circles or closed curves intersect each other if

there are no common elements then they are shown separately as disjoints. Several set

relations can be easily shown by these diagrams. These are useful to illustrate the set

relations, sub as the subset, set relations, and the set-operations such as intersection, union,

complementation, etc. by using regions in a planet to indicate sets. But Venn-diagrams [also

known as Venn Euler diagrams] cannot be used to prove any statement regarding sets. Just asgeometric figure cannot be used to prove geometric theorems. They are more aids for

searching appropriate proofs.


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2.6 Operation of Sets:

2.6.1 Union of Sets: The union of two sets A and V is the set consisting of all elements

which belong to either A or B or Both. The union of A and B

is denoted as AB read as A cup B or A union B.

Symbolically, we have AB = {X: X A or X B or X both A and B}.

In other Words,

X AB => X A or X BFor Example: A = {1, 2}, B = {2, 3}; then AB = {1, 2, 3}

2.6.2 Intersection of Sets: The Intersection of two sets A and B is there set consisting of all

elements which belong to both A and B. The inter-section of

A and B is denoted as AB which is read as A cap B orA intersection BSymbolically, we have

AB = {X: X A and X B}.In other Words,

X AB => X A and X BFor Example: A = {1, 2}, B = {2, 3}; then AB = {2}

2.6.3 Difference of Two Sets: The difference of two sets A

and B is the set of all those elements which belong to A and

not to B and is denoted by A - B (also A ~ B) to be read as

A difference B. Symbolically,

AB = {X : X A and X B}; and BA = {X : X B and X A};

For Example: A = {1, 2}, B = {2, 3}; then A - B = {1}

2.6.4 Complement of Set: The complement of a set is the set of all those elements which do

not belong to that set. In other words, if U be the universalset and A be any set then the complement of set A is the set

UA and is denoted as A, Ac, A or ~A. Symbolically, A = UA

= {X: X U, X A}

={X: X A}

For Example: U = {1, 3, 5, 9, 10, 18}, A = {3, 5, 10};

Then U - A = {1, 9, 18}


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2.7 Symmetric Difference

A difference set is called a symmetric difference of two sets if it contains all those elements

which are in set A and not in set B or those which are in set B and not in set A. they

symmetric difference of two sets A and B will be denoted by

AB = (AB)

(BA)= {X: (X A and X B) or (X B and X A)}

For Example: A = {1, 2, 3, 4, 5, 6}, B= {6, 7, 8}Then, AB = (AB) (BA)

= {1, 2, 3, 4, 5} {7, 8}

= {1, 2, 3, 4, 5, 7, 8}

2.8 Algebra of Sets:

2.8.1 Idempotent Laws (a) A A = A

(b) A A = A2.8.2 Commutative Laws (a) A B = B A

(b) A B = B A2.8.3 Associative Laws (a) (A B) C = A (B C)

(b) (A B) C = A (B C)2.8.4 Distributive Laws (***) (a) A (B C) = (A B) (A C)

(b) A (B C) = (A B) (A C)2.8.5 Identity Laws (a) A = A

(c) A = (b) A U = U

(d) A U = A2.8.6 Complement Laws (a) A A= U

(c) A A= (b) (A)= A(d) AB = A B

2.8.7 DeMorgans Laws (***) (a) (A B)= A B(b) (A B)= AB

2.8.8 Consistency Principle (a) A B if A B = B

(b) A B if A B = A

For Proving Laws, Rules to Remember: X (A B) = X A or X B X (A B) = X A and X B X (A B) = X A and X B X (A B) = X A or X B X (A - B) = X A and X B (X,Y) (A B) = X A or Y B (For Cartesian Product) X A = X A


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2.9 Number of Elements in a Finite Set

n() = 0

n(A) > 0

n(U) = n()n(U) = n(A) + n(A)n(A) = n(AB) + n(AB)n(AB) = n(A) + n(B)n(AB)n(AB C) = n(A) + n(B) + n(C)n(AB) n(BC) n(CA) + n(ABC)

Explanation with Venn diagram

Here,

n(U) = 150

n(A) = 73

n(B) = 51n(C) = 27

n(AB) = 33n(AC) = 15n(BC) = 10n(AB) = 91

n(ABC) = 10

n(ABC) = n(A)+ n(B)+ n(C)- n(AB)- n(BC)- n(AC) + n(ABC) = 103n(ABC) = n(U) - n(AUBUC) = 47n(ABC) = n(AB) - n(ABC) = 23n(ABC) = n(AC) - n(ABC) = 5n(ABC) = n(BC) - n(ABC) = 0n(ABC) = n(A) - n(AB) - n(AC)+ n(ABC) or = 35

= n(A) - n(ABC/) - n(AB/C) - n(ABC)n(ABC) = n(B) - n(AB) - n(BC) + n(ABC) or = 18

= n(B) - n(ABC) - n(ABC) - n(ABC)n(ABC) = n(C) - n(AC) - n(BC) + n(ABC) or = 12

= n(C) - n(ABC) - n(ABC) - n(ABC)n[(AB)C] = n(AC) n(BC) or = 76

n(AC) + n(BC) - n(ABC)

Only two or Exactly two

of the three

n(ABC) +n(ABC) +n(ABC)

[n(ABC) + n(ABC) + n(ABC)]=

n(AB) + n(BC) + n(AC) 3[n(ABC)]

Two or More n(ABC ) +n(ABC) +n(ABC) +n(ABC)

[n(ABC) + n(ABC) + n(A BC) +

n(ABC)]= n(AB) + n(BC) + n(AC) 2[n(ABC)]


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2.10 Cardinality

The cardinality of a set is a measure of the "Distinct number of elements of the set". The

cardinality of a set A is usually denoted |A|, with a vertical bar on each side; this is the samenotation as absolute value and the meaning depends on context. Alternatively, the cardinality

of a set A may be denoted by n(A), A, card (A), or #A.For example, A = {1, 2, 3} contains 3 elements, So |A| = 3 B = {1, 1, 2, 3} contains 3 elements, So |B| = 3

2.11 Cartesian Product:

If A and b be any two sets then the set of all ordered pairs whose first member belongs to set

A and second member belongs to set B is called the Cartesian product of A and b in that

order and is denoted by A B and read as A cross B

In other words, if A, B are two sets, then the set of all ordered pairs of the form (x, y), where

x A and y B is called the Cartesian product of the sets A and B. Symbolically

A B = {(x, y): x A and y B}

For Example:

If A = {1, 2} and B = {a, b}, prove that A B B AFirst, A B = {(1, a), (1, b), (2, a), (2, b)}

Again, B A = {(a, 1), (a, 2), (b, 1), (b, 2)}

So, a B B a (Proven)

2.12 Common notations

i. Z = The Set Of Integers ( -3, -2, -1, 0, 1, 2, 3 )ii. N = The Set Of Nonnegative Integers Or Natural Numbers (1, 2, 3..)

iii. Z+ = The Set Of Positive Integersiv. Q = The Set Of Rational Numbers (-2

, -1

, -

,

, 1

, 2

,)

= Rational Numbers :{A/B|A,Bin integers,BNot Zero}

v. Q+ = The Set Of Positive Rational Numbersvi. Q* = The Set Of Nonzero Rational Numbers

vii. R = The Set Of Real Numbers (-, -e, -2, -,

, 2, -e, -)

viii. R+ = The Set Of Positive Real Numbersix. R* = The Set Of Nonzero Real Numbersx. C = The Set Of Complex Numbers. expressed in the form (a + bi)


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2.13 Problems on Set Theory

1. Foysal, Tania and Mirza are running for chairman of the union council election of Xdistrict. Aftab and Jasim are running for member. How many different tickets of chairman-

member can be formed?

2. Out of the total 152 students in a university, 80 like Debate club, 60 like Language club,20 like both Debate club and Language club. Find the number of students who have not

liked any one of the two.

3. In a survey of 500 students in Business Studies Faculty of XYZ University, 200 werelisted as supporters of Peace club, and 250 were listed as supporters of Music club and 75were stated as the supporters of both the clubs.

i. Find out how many students are only supporter of Pease clubii. Find out how many students are neither supporter of the club

4. In a factory, 60% workers like tea and 30% workers like tea but not coffee. Find thepercentage of workers:

i. Who like both the drinksii. Who like coffee but not tea

5. In a class of 50 students, 30 students have taken statistics, 24 students have taken statistics butnot Mathematics. Find the number of students who have taken Statistics and Mathematics

and those who have taken Mathematics but not Statistics.

6. Two types of defects are observed in a product produced by a manufacturingfirm. The defects are 'X' and 'Y'. Examining 1000 products, the following results are

found:

i. 375 products have defect 'X'ii. 175 products have defect 'X' and Y

iii. How many products are observed with defect Y?7. Among 200 examinees, 40 failed in Mathematics, 20 failed in Statistics and 10 failed in both.

Find the examinee

i. who failed in Mathematics but passed in Statisticsii. who passed only one subject,

iii. who passed at best one subject.8. In a survey of 650 students it was found that 370 students used the college library, 300

had their own and 190 used both college library and their own books.

i. How many students did not use either college library books or their own?ii. How many students used the college library books but not their own books?

iii. How many students used own books but not the college library books?iv. How many students used at least one of the books?

9. Out of 90 students in a class, 10 students do not play any game, 60 students play hockeyand 24 students play basketball only, 14 of them play both the games. Find-

i. How many play basketball?ii. How many play hockeys only?


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10. In an examination, 56% of the students failed in English, 37% failed in Mathematics and17% failed in both the subjects. By using Venn diagram find the percentage of students

who passed in both the subjects.

11. In a survey of 525 readers, it indicates that 350 read newspaper for news, 215listened to radio and 140 watched television. Of the readers, 75 read the newspaper andwatched television, 40 listened radio and watched television and 100 read the newspaper

and listened radio. If 25 used all three sources of news: Find how many readers utilized-

i. at least any one of the threeii. none of the three

12.Out of the total 150 students who appeared for BBA Examination from a college, 45failed in Accounting, 50 failed in Business Mathematics and 30 failed in Statistics. Those

who failed both in Accounting and Business Mathematics were 30, those who failed both

in Business Mathematics and Statistics were 32, and those who failed both in Accounting

and Statistics were 35. The students who failed in all the subjects were 25. Find out thenumber who failed at least any one of the subjects.

13. In a restaurant 90 customers like soup, 48 like rice and 36 like vegetables, 30 like soupand rice both, 24 like vegetable and soup both and if none of them like rice andvegetables both or all the three.

i. Find the number of customers in the restaurant.ii. How many of the customers like only vegetables?

14.Three newspapers named The Independent, The Daily Ittefaq and The Daily Star arepublished in a city. Of the people living in the city, 42% read The Independent, 51% read

The Daily Ittefaq and 68% read The Daily Star. 30% read The Independent and The Daily

Ittefaq, 28% read The Ittefaq and The Daily Star, 36% read The Independent and TheDaily star and 8% read none of these newspapers. What percent read all the three?

15. In a recent survey of 200 women, the following information is found: 122 use Sunsilkshampoo; 98 use All clear shampoo; 74 use Meril shampoo; 52 use Sunsilk and All Clear

shampoo; 42 use Sunsilk and Meril shampoo; 34 use All Clear and Meril shampoo and 26

use all the three.

i. How many do not use any of the three?ii. How many women use only Sunsilk shampoo?

16.Out of the total 84 students in a class, study at least one of the subjects Basic Accounting,Mathematics and Statistics, 28 students study Basic Accounting, 40 statistics and 48Mathematics. 6 students study Basic Accounting and Statistics, 4 Mathematics and Statistics

and there is no students who studies all the three subjects. Find the number of students who

study Basic Accounting and Mathematics but not Statistics.

17. In a survey of a factory, 70%, workers like Fizz Up, 60% like Sprite, 50% Uro Cola, 40%like Fizz Up and Sprite, 30% like Sprite and Uro Cola, 20% like Fizz Up and Uro Cola,

10% like all and 10% like none. The inspector is fined, why?


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18.A group of students of a college conducted a survey on students of BBS who see variousChannels. In the survey it was found that 60% see Boishakhi Channel, 50% see BTV

Channel and 50% see Bangla Vision Channel; 30% see Boishakhi and BTV Channel;

20% see BTV and Bangla Vision Channel; 30% see Bangla Vision and Boishakhi

Channel and 10% do not see any Channel. Find:

i. How many see all the channelsii. How many see exactly two channels

iii. What percentage see Boishakhi channel only19. In a survey concerning soap preferences of consumers, it was found that 50% use Lux, 45%

Aromatic, 40% Keya, 25% Lux and Aromatic, 10% Aromatic and Keya, 16% Keya and

Lux, and 8% all the above three brands. Among the above mentioned soaps, find the

consumer

i. Use at least one,ii. Use none,

iii.

Use only Lux.

20. In a survey concerning the smoking habits of workers of Dhaka city, it was found that 50%smoke Navy cigarette, 45% smoke Gold Leaf cigarette, 40% smoke Benson cigarette, 25%

smoke Navy and Gold Leaf, 10% smoke Gold Leaf and Benson, 16% smoke Benson andNavy, 8% smoke all the three brands. What percentage

i. do not smoke ?ii. smoke only Navy cigarette?

iii. smoke exactly two brands of cigarettes?21.A restaurant has a dessert cart that is brought to each table. At the end of each day, tally is

made of the number of people and the kind of desserts selected. On one day 134 choosefrom the cart; 71 choose ice-cream, 43 choose cake, 61 choose pie, 15 choose pie and cake, 11

choose ice-cream and cake and 3 choose all three. Find:

i. How many picked pie and ice-cream?ii. How many picked exactly two of the items?

iii. How many picked just one item?22. In a survey of 400 families, the number of readers that read recent issues of the

monthly magazine "Ananda Alo" was found to be: March only 72, March but not April 92,March and May 32, March 104, May 192, May and April 32, none of the three months 96 .With the help of set theory find:

i. How many read April issue?ii. How many read only April issue?

iii. How many read two consecutive issues?iv. How many read the May issue, if and only if, they do not read the April issue?v. How many read the March and April issues, but not May issue?

vi. How many read exactly any two months issue?23. In a survey conducted on 2000 workers in a factory it was found that 48% preferred

coffee, 54% liked tea and 64% used to smoke. Of the total, 28% used coffee and tea, 32%

used tea and smoke and 30% preferred coffee and smoke. Only 6% did none of these.

Find the number having (i) all the three, (ii) tea and smoke but not coffee, (iii) Onlycoffee.


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24. In a survey of 100 students, the following results are obtained: 18 students study onEnglish language only, 23 study on English but not Arabic, 8 study on English andBengali, 26 study on English, 48 study on Bengali, 8 study on Bengali and Arabic, 24 do

not study on these three languages. Find:

i. How many students study on English and Arabic but not Bengali?ii. How many students study on Bengali but not Arabic?25.Out of 1000 students who appeared for MBM exam under National University, 750 failed

in Business Mathematics, 600 failed in Management and 600 failed in Micro Economics,

450 failed in both Business Mathematics and Management, 400 failed in both BusinessMathematics and Micro Economics, 150 failed in both Management and Micro

Economics. The students who failed in all the three subjects were 75. Prove that the

above data is not correct.

26.There are three newspapers X, Y and Z published in a city. 60% of people readnewspaper X, 50% of people read newspaper Y, 50% of people read newspaper Z, 30%of people read newspaper X and Y, 20% of people read newspaper Y and Z, 30% of

people read newspaper X and Z, 10% of people do not read any newspapers. Draw aVenn-diagram and find the following:

i. What percent read X, Y and Z?ii. What percent read exactly two newspapers?

iii. What percent read only newspaper X?27.At Tiffin time, 246 students go to college canteen where sells pizza, samucha and chicken

burger. 84 students buy samucha, 72 buy chicken burger and 20 buy only pizza, 30 buysamucha and pizza, 8 buy pizzas and chicken burger but not samucha and 22 buy samucha

and chicken burger but no pizza and 5 buy all the three. Show the information in a Venndiagram and answer the following questions:

i. How many of the students buy nothing at all?ii. How many of the students buy at least two items?

28.Out of the total 60 students appeared for MBA examination of Marketing, Accountingand Management; 25 students failed in Marketing, 24 failed in Accounting, 32 failed in

Management, 9 failed in Marketing alone, 6 failed in Accounting alone, 5 failed in Management

and Accounting only and 3 failed in Marketing and Accounting only. Find:

i. How many failed in all three subjects?ii. How many passed in all the three subjects?

iii. How many failed in marketing and Management but not accounting?iv. How many failed in Management alone?

29. In a Business studies group, all students have taken at least one of the following subjects -Statistics, Geography and English. Of them 130 taken statistics, 140 Geography, 120 English,

50 statistics and Geography, 60 Geography and English, 40 Statistics and English and 20 all

the three. Show the information in a Venn diagram and answer the following questions.

i. How many of the students taken only statistics?ii. How many of the students taken only Geography?

iii. How many of the students taken only English?iv.

How many taken the above subjects?

01. theory of set

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