Set Theory

Sr.noQuestion Option A Option B Option C Option D

Correct

Option

1 Let A={a, b, {c, d}, e}. How many elements does A contain? 1 2 3 4 D

2 Let A={2,{4,5},4} Which statement is correct? 5 is an element of A. {5} is an element of A. {4, 5} is an element of A. {5} is a subset of A. C3 Which of these sets is finite? {x | x is even} ) {1, 2, 3,...} {1, 2, 3,...,999,1000} none C4

Which of these sets is not a null set?A = {x | 6x = 24 and 3x = 1} B = {x | x + 10 = 10}

C = {x | x is a man older than 200 years} D = {x | x < x}

B

6Let A = {x | x is even}, B = {1, 2, 3,..., 99, 100}, C = {3, 5, 7, 9}, D = {101, 102} and E = {101, 103, 105}. Which of these sets can equal S if S A and S and B are disjoint?

A B E C E

7 . Which set S does the power set 2S = {Ф,{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} come from?

{{1},{2},{3}} {1, 2, 3} {{1, 2}, {2, 3}, {1, 3}} {{1, 2, 3}} B

8Let A = {x, y, z}, B = {v, w, x}. Which of the following statements is correct?

A U B ={v, w, x, y, z}

A U B = {v, w, y, z} A U B = {v, w, x, y} A U B ={x, w, x, y, z} A

9 Let U = {1, 2, 3, ..., 8, 9} and A = {1, 3, 5, 7}. Find A'. A' = {2, 4, 6, 8} A' = {2, 4, 6, 8, 9} A' = {2, 4, 6} A' = {9} B

A U B A' A - B B - A

SE Computer 2015 PAT

DM UNIT - 1 MCQs

What is shaded in the Venn diagram below?.

10 C

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Set Theory

11

12 which sets are equal ? 1.{r,s,t} 2.{s,s,t,r} 3.{t,r,t,s} 1 and 2 2 and 3 1 and 3 all are equal D13 A U A=A according to …….law Associative law commutative law Indempotent law distributive law C14 In any application of the theory of sets, the members of all the sets belongs to …… set union intersection universal cardinal C

17 The sets {a,b,c} and {b,c,a} represnet the same sets. TRUE FALSE Both None A18 Cardinality of a set is number of element of the set. TRUE FALSE Both None A19 Multiset is an unordered collection of elemnts where an element can occur a a member

more than onceTRUE FALSE Both None A

20 one of the A or B is uncountably infinite and one is countably infinite then | AUB| will be

countably infinite uncountably finite countably finite uncountably infinite D

21 Sum of first n positive integers is n(n+1) n n(n+1)0.5 n(n+2) C22 Let P(S) denote the power set of set S. which of the is always true P(P(s))=p(s) P(S)∩ S= P(S) P(S)∩P(P(S))={Ф} None C23 The number of elements in the power set P(S) of the set S={{Ф},1,{1,2,3} is 2 4 8 None 826 consider the following data for 120 mathematics students at a college concerning the

languages French,Gernan, and russian: 65 study frensh, 45 study german 42 study russian , 20 study french and german, 25 study french and russian, 15 study german and russian. 8 study all three languages. Determine how many students study exactly 1 subject?

100 25 56 20 C

27 ……… is an unordered collection of elements where an element can occur as a member more than once

Multiset ordered set set None A

28 In a room containing 28 females, there are 18 females who speak English, 15 females speak french and 22 speak german. 9 females speak both english and french, 11 females speak both french and german where as 13 speak both german and english. How many females speak all 3 languages?

9 8 7 6 D

29 If U = {1, 2, 3, . . . 10 } and S = { 4, 5, 6, 7, 8 }, then S ' = { 9, 10 } {1, 2, 3 } {1, 2, 3 9 } {1, 2, 3 9 10 } D

30 If U = {1, 2, 3, . . . 20 } and S = set of prime numbers , then S = { 3, 5, 7, 11, 13, 17 }{ 2, 3, 5, 7, 11, 13, 17, 19 }

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19 }

{1, 2, 3, 5, 7, 11, 13, 17 } B

What is shaded in the Venn diagram below?.

AU B BAA∩B B

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Set Theory

31

37

Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey; 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is

240 340 160 112 C

38

In a town of 10,000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then number of families which buy A only is.

2200 3300 1100 4400 B

39 If A and B are any two sets, then A ∩ (A ∪ B) is equal to B AUB A BUA A40 If a set A has n elements, then the total number of subsets of A is. n(n+1) 2n n*n 3n C41 If A and B are any two sets, then A ∪ (A ∩ B) is equal to. B A AUB BUA B

42 The set of intelligent students in a class is An ull set Singleton set A finite setNot a well defined

collectionD

43 The number of proper subsets of the set {1, 2, 3} is. 8 4 6 2 C

In the Venn diagram opposite, the shaded portion represents.

In the figure opposite, U = { a, b, c, d, e, f, g, h }. (X∩Y) U Z =

D

{f}

P U Q P∩ Q' P' ∩ Q P∩ Q

32 {c,e,f,g,h} {e,f,g}

B

{c,f,h} A

C34 (AUC) ∩ B (BUA)∩ C (A U C)' A' U C'

In the figure below, the shaded portion represents?

(Y∩ Z) U X33 (X∩ Z) U Y (X∩Y)U Z (Z∩ Y) U X

In the figure below, the shaded portion represents?

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Set Theory

44 The smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9} is {2,3,5} {3,5,9} {1,2,3,5} {1,8,9} B45 If A, B, C be three sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then. A=B A=B=C B=C A=C C46 If A, B and C are any three sets, then A - (B ∪ C) is equal to (A-B)U(A-C) (A - B) ∩ (A - C) A-B A-C B47 If A, B and C are any three sets, then A × (B ∪ C) is equal to (A × B) ∪ (A × C) (A × B) ∩ (A × C) (A ∪ B) × (A ∪ C) None A48 A = {x: x ≠ x }represents. {} {0} {1} {x} A49 Which of the following is a subset of {b , c , d }? {a} {} {1} {2} B50 How many subsets does the set {a , b , c , d , e } have? 8 5 32 4 C

51

Some girls play netball. Tall girls are over 1.8 meters in height. All netball players are tall girls. Lee is 1.9 meters tall and does not play netball. Which Venn diagram below represents the statements above? U= Girls P= Plays Netball D= Does Not Play T= Tall Girls

B

52Consider the statement form p ) q where p =“If Tom is Jane’s father then Jane is

Bill’s niece” and q =“Bill is Tom’s brother.” Which of the following statements isequivalent to this statement?

If Bill is Tom’s Brother, then Tom is

Jane’s father and Jane is not Bill’s n

If Bill is not Tom’s Brother, then Tom is Jane’s father and Jane

is not Bill’s niece.

If Bill is not Tom’s Brother, then Tom is

Jane’s father or Jane is Bill’s niece.

If Bill is Tom’s Brother, then Tom is Jane’s father and Jane

is Bill’s niece.

B

53Consider the statement,“If n is divisible by 30 then n is divisible by 2 and by 3 andby 5.”Which of the following statements is equivalent to this statement?

If n is not divisible by 30 then n is divisible by 2 or divisible by 3 or divisible by 5

If n is not divisible by 30 then n is not divisible by 2 or not divisible by 3 or not divisible by 5

If n is divisible by 2 and divisible by 3 and

divisible by 5 then n is divisible by 30.

If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30

D

54Which of the following statements is the contrapositive of the statement, “You win thegame if you know the rules but are not overconfident.”

If you lose the game then you don’t know the rules or you are overconfident.

A sufficient condition that you win the game is that you know the rules or you are not over confident

If you don’t know the rules or are overconfident

you lose the game.

If you know the rules and are overconfident then you win the game.

A

55 The statement form (p↔r)→(q↔r) is equivalent to.

56A sufficient condition that a triangle T be a right triangle is that a2 + b2 = c2. An equivalent statement is

If T is a right triangle then a2 + b2 = c2.

If a2 + b2 = c2 then T is a right triangle.

If a2 + b2 6= c2 then T is not a right triangle.

T is a right triangle only if a2 + b2 = c2.

B

57Which of the following is the inverse of the statement: " If I eat a mango than I do not drink milk".

I drink milk only if I do not eat a mango

If I don’t eat a mango then I drink milk

If I do not drink milk then I eat mango

None B

58If p= It is hot, and q= It is sultry, which of the following sentences in the appropriate version for the symbolic expression: -p٨ q

If it is sultry then it is hot

It is sultry only if it is hot

It is sultry and it is not hot

None D

59which of the following is the contrapositive of the statement: " A quadrilateral is a square only if it is both rectangle and a rhombus".

If a rectangle is not a a rhombus it is not a square

If a rhombus is not rectangle it is not a square

If a quadrilateral is neither a rectangle nor a rhombus then it is not a square

None C

60 For a conditional statement p===>q, which of the following is incorrect.Converse of the inverse is its contrapositive

contrapositive of the converse is its inverse

Inverse of the contrapositive is its converse

None D

61 Which of the following is equivalent to p==>q ~xp٨q pV~q ~xpVq pV~q C62 Equivalent inverse of p==>q is ~xp٨q pV~q ~xpVq pV~q D63 Converse of p==>q is ~xp٨q pV~q ~xpVq pV~q D64 the truth table for exclusive disjunction will be tautology Contradiction Logical equivalent p or q but not both D65 The conditional statement p→q and its contrapositive are…. Converse Inverse Logically equivalent None C

66Set A has 3 elements, set B has 6 elements, then the minimum number of elements in A U B is ….

3 6 9 None B

67Set A has 3 elements, set B has 6 elements, then the minimum number of elements in A U B is ….

6 9 18 None B

68 if P∩ Q= Ф then P U Q' is P U-P U-Q Ф C

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Set Theory

69 The number of proper subset of {1,2,3,4} is 16 15 10 12 A

70Consider the four statements: 1.(q==>p)٨(~p) 2. p==>(~q) V r 3. ~p==>~(p٨q) 4.p٨q٨~(pVq) Which one is tautology.

A B C D C

71 Which among the statements given in Q.70 is contradiction A B C D D

72Consider the four statements 1.p٨~q↔~p V q 2.p٨(~p Vq)٨(~q) 3.(pV(~q))٨ ~((~p)Vq) 4.~(p↔q)↔(p↔~q) which one of this contradiction.

A B C D D

73consider the four tsatements: 1.(p→q) ٨(p٨~q) 2.(~p→r)٨(p↔q) 3.p→(~qVr) 4.~(p٨q) V (p↔q) Which one of these four ststements is a tautology.

A B C D D

74 In above question which statement is contradiction. A B C D A

75Which of the following sets are equal. 1. {p,q,m,n} 2.{m,p,n,q} 3.{q,p,p,m,m,p,n} 4.{p,q,n,,n,m}

1 and 2 are equal 2 and 3 are equal 3 and 4 are equal All are equal. D

76 Consider the set A={{1,3,5},{7,9,11},{13,15}} then determine which of the following is/are true. 1.1ЄA 2.{{1,3,5}} CA 3. Ф subet of A 4. A

2 and 3 is true 1 and 3 is true 3 is true None A

77Determine the validity of the following argument: S1: all my friends are musicians. S2: John is my friend. S3: None of my neighbours are musicians. S: John is not my neighbour.

Valid Not valid Both a and b None A

78

In a survey of 60 people , it was found that: 25 read Newsweek magzine. 26 read Time 26 read Fortune 9 read both newsweek and fortune 11 read both Newsweek and Time 8 read both Time and Fortune 3 read all 3 magzines. 1. Find the number of people who read at least one of the three magzines

30 52 40 68 B

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Set Theory

79 In above Q.78 Find the number of people who read exactly 1 magzine. 30 52 40 68 A

80

In a class of 80 students , 50 students know English, 55 know french and 46 know german language. 37 students know english and french, 28 students know french and german, 7 students know none of the languages. Find out how many students know all the three languages?

73 72 50 54 A

81 In above q.80 how many students exactly know 2 languages? 52 54 60 25 B82 In q. 80 how many students know exactly 1 language? 54 12 7 8 C83 A preposition is a statement that is either ture or false TRUE FALSE none both a and b A

84 A prepostition that is true under all circumstances is referred to as a …. Tautology Contradiction Negation Sentence A85 A prepostition that is false under all circumstances is referred to as a …. Tautology Contradiction Negation Sentence B86 p→q is logically equivalent to ~p V q according to… Identity law Implication law associative law Absoption law B

87A logical expression which consist of a product of elementary sum is caleed…..

Disjunctive normal form

Conjunctive normal form

Normal form None B

88A logical expression which consist of a sum of product is caleed…..

Disjunctive normal form

Conjunctive normal form

Normal form None A

89 An assertion that contains one or more variable is called a…. CNF DNF Predicates Quantifiers C

90Determine the validity of argument given: s1: If I like mathematics then I will study. S2: Either I will study or I will fail. S: If I fail then I do not mlike mathematics.

Valid Invalid Both a and b none B

91 p V ~(p٨q) is…. Contradiction Tautology predicate None B

92Determine the validity of the argument s1: If I stay up late at night , then I will be tired in the morning. S2: I stayed up last last night s: I am tired this morning.

Valid Invalid Both a and b none A

93An argument is valid if, whenever the conclusion is true, thenthe premises are also true.

TRUE FALSE both a and b none B

94 De Morgan's laaws are two examplesof rules of inference TRUE FALSE both a and b none B

95In a club , all members participate either in tambola or the fete. 420 participate in the fete, 350 play tambola and 220 participate in both. How many members does the club have?

250 550 120 140 B

96 dual of (p V q)٨ r is.. p Vq (p٨q) Vr p ٨r (p Vq) Vr B

97

It was found that in first year of computer science of 80 students 50 know Cobol, 55 know C, 46 know pascal. It was also known that 37 know C and cobol, 28 know C and pascal , 25 know pascal and cobol, 7 students know none of the languages. Find how many all the 3 languages?

10 12 35 9 B

98 In above q.97 How many know exactly 2 languages? 54 16 10 35 A99 In q.97. How many know exactly 1 language? 6 16 7 10 C

100

In the class of 55 students the number ofstudying different subjects are as given below: Maths 23, Physics 24, chemistry 19, maths+physics 12, maths+chemistry 9, Physics +chemistry 7, all three subjects 4. Find the number of students who have taken atleast 1 subject?

22 45 42 14 C

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Set Theory

101 [~ q ^ (p→q)]→~ p is, Satisfiable tautology unsatisfiable contradiction B

102

If P and Q stands for the statementP : It is hotQ : It is humid,then what does the following mean?P Ù (~ Q):

It is got and it is humid

It is hot and it is not humid

it is not hot and it is humid none B

103

In a survey of 85 people it is found that 31 like to drink milk, 43 like coffee and 39 like tea.Also 13 like both milk and tea, 15 like milk and coffee, 20 like tea and coffee and 12 like none of the three drinks. Find the number of people who like all the three drinks.

10 9 8 7 c

104 The statement ( p^q) → p is a absurdity contadiction tautology none c

105Let p be “He is tall” and let q “He is handsome”. Then the statement “It is false that he is short or handsome” is:

p ^ q ~ (~ p ^q) p^ ~ q ~ p ^q B

106 Let P(S) denotes the powerset of set S. Which of the following is always true? P(P(S)) = P(S) P(S) IS = P(S) P(S) I P(P(S)) = {ø} S € P(S) D107 Which of the following proposition is a tautology? (p v q)→p p v (q→p) p v (p→q) p→(p→q) C

108Which of the following statement is the negation of the statement “4 is even or -5 is negative”?

4 is odd and -5 is not negative

4 is even or -5 is not negative

4 is odd or -5 is not negative

4 is even and -5 is not negative

A

109 Which one is the contrapositive of q → p ? p → q ~p →~q ~q→~p None b

110

Check the validity of the following argument :- “If the labour market is perfect then the wages of all persons in a particular employmentwill be equal. But it is always the case that wages for such persons are not equaltherefore the labour market is not perfect.”

Invalid Valid Both a and b None B

111∃ is used in predicate calculusto indicate that a predicate is true for all members of a specified set.

TRUE FALSE Both a and b None A

112∀ is used in predicate calculusto indicate that a predicate is true for at least one member of a specified set.

TRUE FALSE Both a and b None A

113 “If the sky is cloudy then it will rain and it will not rain” absurdity contadiction tautology none c

114Represent statement into predicate calculus forms : "Not all birds can fly". Let us assume the following predicates bird(x): “x is bird” fly(x): “x can fly”.

∃x bird(x) V fly(x) ∃x bird(x) ^ ~ fly(x) ∃x bird(x) ^ fly(x) None B

115

Represent statement into predicate calculus forms : "If x is a man, then x is a giant." Let us assume the following predicatesman(x): “x is Man”giant(x): “x is giant”.

∀ (man(x)→ ~giant(x))

∀ man(x)→ giant(x) ∀ (man(x)→ giant(x)) None C

116

Represent statement into predicate calculus forms : "Some men are not giants." Let us assume the following predicatesman(x): “x is Man”giant(x): “x is giant”.

∃x man(x) ^ giant(x) ∃x man(x) ^ ~ giant(x) ∃x man(x) V ~ giant(x) None B

117

Represent statement into predicate calculus forms : There is a student who likes mathematics but not history. Let us assume the following predicatesstudent(x): “x is student.”likes(x, y): “x likes y”. and ~likes(x, y) “x does not like y”.

∃x [student(x) ^ likes(x, mathematics) ^~ likes(x, history)]Q.

∃x [student(x) ^Vlikes(x,

mathematics) V~ likes(x, history)]Q.

∃x [student(x) ^ ~likes(x, mathematics) ^likes(x,

history)]Q.None A

118 AUB = (A− B)U(B−A)U(AпB). FALSE TRUE Both a and b None B119 [(PVQ)^(P→R)^(Q→S)] → (SVR). Is a…. absurdity contadiction tautology none c120 ~(x vy) = ~x ^ ~y FALSE TRUE Both a and b None B121 (x ^ y)’ = x’ V y’ FALSE TRUE Both a and b None B

122Test the validity of argument:“If it rains tomorrow, I will carry my umbrella, if its cloth is mended. It will rain tomorrow and the cloth will not be mended. Therefore I will not carry my umbrella”

Invalid Valid Both a and b None B

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Set Theory

123

In a group of athletic teams in a certain institute, 21 are in the basket ball team, 26 in the hockey team, 29 in the foot ball team. If 14 play hockey and basketball, 12 play foot ball and basket ball, 15 play hockey and foot ball, 8 play all the three games. (i) How many players are there in all?

78 98 23 43 D

124 In above Q.123 (ii) How many play only foot ball? 10 8 9 4 A125 (p ↔ q) ↔ r = p ↔ (q ↔ r) absurdity contadiction tautology none c

126 Write the negation in good english sentence : "Jack did not eat fat, but he did eat broccoli."

If Jack eat and broccoli then he did

ate fat.

If Jack did not eat broccoli then he did

ate fat.

If Jack did not eat broccoli or he did ate fat.

If Jack did not eat broccoli then he did

not ate fat.B

127 Write the negation in good english sentence : The weather is bad and I will not go to work.

The weather is not bad or I will go to

work.

The weather is good or I will go to work.

The weather is not bad or I will not go to work.

None A

128 Write the negation in good english sentence : Mary lost her lamb or the wolf ate the lamb.

Mary did loss her lamb and the wolf

eat the lamb.

Mary did loss her lamb and the wolf did

not eat the lamb.

Mary did not loss her lamb and the wolf did not

eat the lamb.None C

129 Write the negation in good english sentence : I will not win the game or I will not enter the contest.

I will not win the game and I will enter

the contest.

I will win the game and I will enter the contest.

I will win the game and I will not enter the contest.

None B

130

In a survey of 85 people it is found that 31 like to drink milk 43 like coffee and 39 like tea.Also 13 like both milk and tea, 15 like milk and coffee, 20 like tea and coffee and 12 like none of the three drinks. Find the number of people who like all the three drinks.

9 8 10 11 B

131Find the negation of the proposition: “Michael’s PC runs Linux”

“It is not the case that Michael’s PC runs Linux.”

“Michael’s PC does not run Linux.”

Both a and b Only a C

132A proof that begins by asserting a claim and proceeds to show that the claim cannot be true is by

Induction Contradiction prevarication construction B

133 A proof that proceeds by showing the existence of something desired is by Induction Contradiction prevarication construction A

134Proofs by contradiction

dismiss certain rules of logic misrepresent facts

start by assuming the opposite of what is to be proven

end by rejecting what is to be proven

C

135 Induction is a algorithm program Proof Proof method D136 ^ denotes union AND set membership negation b137 ~ denotes union AND set membership negation D138 The induction principle makes assertions about infinite sets large finite sets small finite sets logical formulas139 Quantifiers ____ variables Negate Change give values to bind C

140A validity-maintaining procedure for deriving sentences in logic from other sentences is Proof Theorem Inference rule inference chain

C

141 Inference rules maintain completeness validity satisfiablity logic B

142A validity-maintaining procedure for deriving sentences in logic from other sentences is a Proof Theorem Inference rule inference chain

C

143Predicate logic is algorithm language of assertions

language of arithmetic expressions set of symbols

144 If A and B be sets and AC and Bc denote the complements of the sets A and B, then set

(A — B) ∪ (B — A) ∪ (A ∩ B) is equal to Ac ∪ Bc Ac ∩ Bc A ∪ B A ∩ BC

145 Number of proper subsets of a set of order three 3 6 8 9 B146 If A be a finite set of size n, then number of elements in the power set of A x A is 22^n 2n^2 (2n)2 none B

147 Which of the following set (s) are empty ? {x : x = x} {x : x ≠ x} {x : x = x2} {x : x ≠ x2} B

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Set Theory

148Which of the following sets is a null set ? I. X = {x | x= 9, 2x = 4 } II. Y = {x | x= 2x.x ≠ 0 } III. Z = { x | x-8 = 4 } 1 and 2 only 1 , 2 and 3 2 and 3 1 and 3

A

149Determine the validity of argument: " All guilty people will be arrested. All thieves are guilty people. Therefore, all thieves will be arrested. Valid Invalid Both a and b None

A

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Set Theory

150Which of the following diagrams indicates the best relation between Women, Mothers and Engineers ?

A

151Which of the following diagrams indicates the best relation between Profit, Dividend and Bonus ?

B

152Which of the following diagrams indicates the best relation between Travelers, Train and Bus ?

C

153Which of the following diagrams indicates the best relation between Factory, Product and Machinery ?

D

154Which of the following diagrams indicates the best relation between Author, Lawyer and Singer ?

B

155Which of the following diagrams indicates the best relation between Judge, Thieves and Criminals ?

B

156Which of the following diagrams indicates the best relation between India, Haryana and World ?

D

157Which of the following diagrams indicates the best relation between Pigeon, Bird and Dog ?

A

158Which of the following diagrams indicates the best relation between Earth, Sea and Sun ?

A

159Which of the following diagrams indicates the best relation between Hockey, Football and Cricket ?

B

160Which of the following diagrams indicates the best relation between Examination, Questions and Practice ?

c

161Which of the following diagrams indicates the best relation between Bulb, Lamp and Light ?

C

162Which of the following diagrams indicates the best relation between Lion, Dog and Snake ?

C

163Which of the following diagrams indicates the best relation between Hospital, Nurse and Patient ?

C

164n a Venn diagram , the overlap between two circles represents: the union of two sets

the intersection of two sets

the elements that are in either of two sets

the difference between the number of elements in two sets

B

165Which of these subsets are equal: A = {r.t.s} B = {s,t,r,s} C = {t,s,t,r} D = {s,r,s,t} A and B A and C B and D all are equal

D

166 Determine the total number of subsets of the following set: {h,i, j, k, l, m, n} 128 64 32 14 A

168If B is a Boolean Algebra, then which of the following is true

B is a finite but not complemented lattice

B is a finite, complemented and distributive lattice

B is a finite, distributive but not complemented lattice

B is not distributive lattice

B

169 The statement ( p^q) _ p is a Contingency contradiction tautology None C

170

In a computer laboratory out of 6 computers. A) 2 have floating point arithmatic unit. B) 5 have magnetic disk memory. C) 3 have graphics display. D) 2 have both floating point arithmatic unit and magnetic disk memory. E) 3 have both magnetic disk memory and graphics display F) 1 has both floating point arithmatic unit and graphics display. G) 1 has all the three specifications. How many have atleast 1 specification?.

6 5 4 2 5

171

1. Let m = “Juan is a math major,” c = “Juan is a computer science major,” g = “Juan’s girlfriend is a literature major,” h = “Juan’s girlfriend has read Hamlet,” and t = “Juan’s girlfriend has read The Tempest.” Which of the following expresses the statement “Juan is a computer science major and a math major, but his girlfriend is a literature major who hasn’t read both The Tempest and Hamlet.”

c ∧ m ∧ (g ∨ (∼h ∨ ∼t))

c ∧ m ∧ g ∧ (∼h ∧ ∼t) c ∧ m ∧ g ∧ (∼h ∨ ∼t) c ∧ m ∧ (g ∨ (∼h ∧ ∼t))

C

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Set Theory

172 3. The truth table for (p ∨ q) ∨ (p ∧ r) is the same as the truth table for (p ∨ q) ∧ (p ∨ r) (p ∨ q) ∧ r (p ∨ q) ∧ (p ∧ r) p V q D

1736. Consider the statement, “Either −2 ≤ x ≤ −1 or 1 ≤ x ≤ 2.” The negation of this statement is

x < −2 or 2 < x or −1 < x < 1 (x < −2 or 2 < x −1 < x < 1

x ≤ −2 or 2 ≤ x or −1 < x < 1

A

174

8. Which of the following statements is FALSE:

(P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to ∼Q ∧ ∼P

(P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to Q ∨ P

(P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to Q ∨ (P ∧ ∼Q)

(P ∧ Q) ∨ (∼P ∧ Q) ∨ (P ∧ ∼Q) is equal to [(P ∨ ∼P) ∧ Q] ∨ (P ∧ ∼Q)

A

175anya is older than Eric. Cliff is older than Tanya. Eric is older than Cliff. If the first two statements are true, the third statement is TRUE FALSE Both None

B

176Blueberries cost more than strawberries. Blueberries cost less than raspberries.Raspberries cost more than both strawberries and blueberries. If the first two statements are true, the third statement is TRUE FALSE Both None

A

177All the trees in the park are flowering trees. Some of the trees in the park are dogwoods. All dogwoods in the park are flowering trees. If the first two statements are true, the third statement is TRUE FALSE Both None

A

178Mara runs faster than Gail.Lily runs faster than Mara.Gail runs faster than Lily.If the first two statements are true, the third statement is TRUE FALSE Both None

B

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Set Theory

179

Apartments in the Riverdale Manor cost less than apartments in The Gaslight Commons.Apartments in the Livingston Gate cost more than apartments in the The Gaslight Commons.Of the three apartment buildings, the Livingston Gate costs the most.If the first two statements are true, the third statement is TRUE FALSE Both None

A

180 The power set of the set {ϕ} is {ϕ} {ϕ, {ϕ}} {0} None B

181Fact 1:All dogs like to run.Fact 2:Some dogs like to swim.Fact 3:Some dogs look like their masters.If the first three statements are facts, which of the following statements must also be a fact? I:All dogs who like to swim look like their masters. II:Dogs who like to swim also like to run.III:Dogs who like to run do not look like their masters. I only II only III only All

B

182

Fact 1: Jessica has four childrenFact 2: Two of the children have blue eyes and two of the children have brown eyes.Fact 3: Half of the children are girls.If the first three statements are facts, which of the following statements must also be a fact?I: At least one girl has blue eyes.II: Two of the children are boys.III: The boys have brown eyes. I only II only III only All

B

183

Fact 1: All drink mixes are beverages.Fact 2: All beverages are drinkable.Fact 3: Some beverages are red.If the first three statements are facts, which of the following statements must also be a fact?I: Some drink mixes are red.II: All beverages are drink mixes.III: All red drink mixes are drinkable. I only II only III only All

C

184

Fact 1: All chickens are birds.Fact 2: Some chickens are hens.Fact 3: Female birds lay eggs.If the first three statements are facts, which of the following statements must also be a fact?I: All birds lay eggs.II: Some Hens are birds.III: Some chickens are not hens. I only II only II and III only All

C

185100 sportsmen were asked whether they play which game: Cricket, hockey,Football. The results were : 45 play cricket, 38 play hockey, 21 play football, 18 play cricket and hockey, 9 play cricket and football, 4 play football and hockey and 23 play none of these. Determine the number of sportsmen who play exactly 1game 54 84 56 78

A

186 In above Ex. 196 how many players play exactly 2 of the games? 29 79 19 39 C187 A theory of sets was firstly introduced by…. Tim berners lee Franklin G.Canter c.panther C

188the inventor of set defined set as

collection od distinct objects collection of memebers

simple as collection of objects

simply as collection of alphabets

C

189 class, groups , collection are synonyms of the term set. TRUE FALSE both a and b none A1 The set O of odd positive integers less than 10 can be expressed by {1, 2, 3} {1, 3, 5, 7, 9} {1, 2, 5, 9} {1, 5, 7, 9, 11} B2 Power set of empty set has exactly _____ subset 1 2 0 3 A3 For a finite set A with „n‟ elements, The power set P(A) contains .............elements. 2^n 2^(n-1) 2^(n+1) None of these A

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Set Theory

4 There are 8 students on the curling team and 12 students on the badminton team. What is the total number of students on the two teams if three students are on both teams:

20 17 15 14B

5 In a Venn diagram , the overlap between two circles represents: the union of two sets the intersection of two sets

the elements that are in either of two sets

the difference between the number of elements in two sets

B

6 In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, those whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

19 41 21 57

B

7 Of the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone. 40 of them had both, a two-wheeler and a credit card, 30 had both, a credit card and a mobile phone and 60 had both, a two wheeler and mobile phone and 10 had all three. How many candidates had none of the three?

0 20 10 18

C

8 In a class 40% of the students enrolled for Math and 70% enrolled for Economics. If 15% of the students enrolled for both Math and Economics, what % of the students of the class did not enroll for either of the two subjects?

5% 12% 10% 6%A

9 In a class of 40 students, 12 enrolled for both English and German. 22 enrolled for German. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only English and not German?

10 30 18 20C

10 Which statement represents "all numbers between negative 4 and positive 8" ? -4 < 8 -4 < x < 8 0 -3

1. A _______ is an ordered collection of objects.

a) Relation

b) Function

c) Set

d) Proposition

Answer: c

Explanation: By the definition of set.

2. The set O of odd positive integers less than 10 can be expressed by ___________ .

a) {1, 2, 3}

b) {1, 3, 5, 7, 9}

c) {1, 2, 5, 9}

d) {1, 5, 7, 9, 11}

Answer: b

Explanation: Odd numbers less than 10 is {1, 3, 5, 7, 9}.

3. Power set of empty set has exactly _____ subset.

a) One

b) Two

c) Zero

d) Three

Answer: a

Explanation: Power set of null set has exactly one subset which is empty set.

4. What is the Cartesian product of A = {1, 2} and B = {a, b}?

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

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

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

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

Answer: c

Explanation: A subset R of the Cartesian product A x B is a relation from the set A to the set B.

5. The Cartesian Product B x A is equal to the Cartesian product A x B. Is it True or False?

a) True

b) False

Answer: b

Explanation: Let A = {1, 2} and B = {a, b}. The Cartesian product A x B = {(1, a), (1, b), (2, a),

(2, b)} and the Cartesian product B x A = {(a, 1), (a, 2), (b, 1), (b, 2)}. This is not equal to A x B.

6. What is the cardinality of the set of odd positive integers less than 10?

a) 10

b) 5

c) 3

d) 20

Answer: b

Explanation: Set S of odd positive an odd integer less than 10 is {1, 3, 5, 7, 9}. Then, Cardinality

of set S = |S| which is 5.

7. Which of the following two sets are equal?

a) A = {1, 2} and B = {1}

b) A = {1, 2} and B = {1, 2, 3}

c) A = {1, 2, 3} and B = {2, 1, 3}

d) A = {1, 2, 4} and B = {1, 2, 3}

Answer: c

Explanation: Two set are equal if and only if they have the same elements.

8. The set of positive integers is _________ .

a) Infinite

b) Finite

c) Subset

d) Empty

Answer: a

Explanation: The set of positive integers is not finite.

9. What is the Cardinality of the Power set of the set {0, 1, 2}.

a) 8

b) 6

c) 7

d) 9

Answer: a

Explanation: Power set P ({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,P({0, 1, 2}) =

{null , {0}, {1}, {2}, {0, 1}, {0,2}, {1, 2}, {0, 1, 2}}.

10. The members of the set S = {x | x is the square of an integer and x < 100} is

_________________. a) {0, 2, 4, 5, 9, 58, 49, 56, 99, 12} b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81} c)

{1, 4, 9, 16, 25, 36, 64, 81, 85, 99} d) {0, 1, 4, 9, 16, 25, 36, 49, 64, 121

Answer: b

Explanation: The set S consists of the square of an integer less than 10.

1. Let A and B be two sets in the same universal set. Then A B =

(a) A B

(b) A B

(c) A B

(d) None of these

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

(a) n

(b) 2n 1

(c) n2

(d) 2n

3. For any two sets A and B, A (A B) =

(a) A

(b) B

(c)

(d) None of these

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

(a) 4 A

(b) {4} A

(c) B A

(d) None of these

5. The symmetric difference of A and B is

(a) (A B) (B A)

(b) (A B) (B A)

(c) (A B) (A B)

(d) {(A B) A} {(A B) B)

6. The symmetric difference of A = {1, 2, 3} and B = {3, 4, 5} is

(a) {1, 2}

(b) {1, 2, 4, 5}

(c) {4, 3}

(d) {2, 5, 1, 4, 3}

7. For any two sets A and B, (A B) (B A) =

(a) (A B) A

(b) (B A) B

(c) (A B) (A B)

(d) (A B) (A B)

8. Which of the following statement is false:

(a) A B = A B

(b) A B = A (A B)

(c) A B = A B

(d) A B = (A B) B

9. For any three sets A, B and C

(a) A (B C) = (A B) (A C)

(b) A (B C) = (A B) C

(c) A (B C) = (A B) (A C)

(d) A (B C) = (A B) (A C)

10. Let A = {x | x R, x 4} and B = {x R | x < 5}. Then A B =

(a) (4, 5)

(b) (4, 5)

(c) (4, 5)

(d) (4, 5)

11. Let U be the universal set containing 700 elements. If A, B are sub-sets of U such

that n(A) = 200, n(B) = 300 and n (A B) = 100. Then n (A B) =

(a) 400

(b) 600

(c) 300

(d) None of these

12. If A = {1, 2} and B = {0, 1}, then A B is

(a) {(1, 0) (1, 1), (2, 0) (2, 1)}

(b) {(1, 0), (2, 1)}

(c) {(1, 1), (1, 2), (0, 1) (0, 2)

(d) None of these

13. If A = {1, 2, 3}, B = {3, 4, 5}, then (A B) A is

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

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

(c) {(1, 3), (3, 1), (3, 2)

(d) None of these

14. If A = {x R | 0 < x < 1} and B = {x R | 1 < x < 1, then A the set

(a) Of all points lying inside the rectangle having vertices at (1, 1), (0, 1), (0, 1) and (1,

1)

(b) Of all points lying inside the rectangle having vertices at (1, 0), (1, 1), (0, 1) and (0, 0)

(c) Of all points lying on the sides of the rectangle whose vertices are at (1, 1), (0, 1), (0,

1) and (1, 1)

(d) None of these

15. If A = {2, 3, 4}, B = {2, 5, 6}, then (A B) (A B) is

(a) {(3, 2), (3, 3), (3, 5)

(b) {(3, 2), (3, 5) (3, 6)

(c) {(3, 2), (3, 5)}

(d) None of these.

16. If A = {1, 2, 3}, B = {4, 5, 6} and C = {1, 2}, then (A B) (A C) is

(a) {(1, 3), (1, 5)}

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

(c) {(1, 2), (1, 3), (1, 5)}

(d) None of these

17. If A = {1, 2}, B = {2, 3} and C = {4}, then A B C is

(a) {(1, 2, 4), (2, 2, 4), (1, 3, 4), (2, 3, 4)

(b) {(1, 2, 4), (1, 4, 3), (2, 3, 4)

(c) {(1, 3, 4), (2, 3, 4), (2, 1, 3), (2, 2, 4)

(d) None of these

18. If A = {a, b, c}, B = }c, d, e}, C = {a, d, f}, then A (B C) is

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

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

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

(d) None of these.

19. If A = {2, 3} and B = {x | x N and x < 3}, then A B is

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

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

(c) {(1, 2), (2, 2), (3, 3), (3, 2)

(d) None of these.

20. If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, then (A B) (B C) is

(a) {(1, 2), (1, 5), (2, 5)}

(b) {(1, 4)}

(c) (1, 4)

(d) None of these.

Answer Keys :

1. c 2. d 3. a 4. d 5. b

6. b 7. c 8. c 9. abc 10. c

11. c 12. a 13. b 14. a 15. c

16. d 17. a 18. d 19. a 20. b

1. The relation R defined in A = {1, 2, 3} by aRb, if | a2 – b2 | 5. Which of the following

is false?

(a) R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}

(b) R–1 = R

(c) Domain of R = {1, 2, 3}

(d) Range of R = {5}

2. The relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(x, y) : | x2 – y2| < 16} is

given by

(a) {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}

(b) {(2, 2), (3, 2), (4, 2), (2, 4)}

© {(3, 3), (4, 3), (5, 4), (3, 4)}

(d) None of the above

3. If R = {x, y) : x, y Z, x2 + y2 4} is a relation in z, then domain of R is

(a) {0, 1, 2}

(b) {– 2, – 1, 0}

(c) {– 2, – 1, 0, 1, 2}

(d) None of these

4. If A = { (1, 2, 3}, then the relation R = {(2, 3)} in A is

(a) symmetric and transitive only

(b) symmetric only

(c) transitive only

(d) not transitive

5. Let X be a family of sets and R be a relation in X, defined by ‘A is disjoint from B’.

Then, R is

(a) reflexive

(b) symmetric

(c) anti-symmetric

(d) transitive

6. R is a relation defined in Z by aRb if and only if ab 0, then R is

(a) reflexive

(b) symmetric

(c) transitive

(d) equivalence

7. Let a relation R in the set R of real numbers be defined as (a, b) R if and only if 1 +

ab > 0 for all a, bR. The relation R is

(a) Reflexive and Symmetric

(b) Symmetric and Transitive

(c) Only transitive

(d) An equivalence relation

8. If R be relation ‘

(a) {(1, 4), (2, 5), (3, 6), ….}

(b) { (4, 1), (5, 2), (6, 3), ….}

(c) {(4, 1), (5, 2), (6, 3), ….}

(d) None of the above

13. Two finite sets A and B have m and n elements respectively. If the total number of

subsets of A is 112 more than the total number of subsets of B, then the value of m is

(a) 7

(b) 9

(c) 10

(d) 12

14. Let X and Y be the sets of all positive divisors of 400 and 1000 respectively

(including 1 and the number). Then, n (X Y) is equal to

(a) 4

(b) 6

(c) 8

(d) 12

Answer keys:

1. d 2. d 3. c 4. d 5. b

6. d 7. a 8. c 9. c 10. b

11. d 12. b 13. a 14. D

MCQs: Set Theory & Venn Diagrams

Exercise

If A and B are two sets, A â?© B represents:

a) all elements in either A and B

b) all elements in both A and B

c) all elements that are in A but not B

d) all sets that include A and B

There are 8 students on the curling team and 12 students on the badminton team. What is the

total number of students on the two teams if three students are on both teams:

a) 20

b) 17

c) 15

d) 14

Determine the total number of subsets of the following set: {h,i, j, k, l, m, n}

a) 128

b) 64

c) 32

d) 14

In a Venn diagram , the overlap between two circles represents:

a) the union of two sets

b) the intersection of two sets

c) the elements that are in either of two sets

d) the difference between the number of elements in two sets

Which of these subsets are equal:

A = {r.t.s} B = {s,t,r,s} C = {t,s,t,r} D = {s,r,s,t}

a) A and B

b) A and C

c) B and D

d) all are equal