210+ Discrete Structure (DS) Solved MCQs

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Chapter: Unit 1
1.

Let A={a, b, {c, d}, e}. How many elements does A contain?

A. 1
B. 2
C. 3
D. 4
Answer» D. 4
discuss
2.

Let A={2,{4,5},4} Which statement is correct?

A. 5 is an element of A.
B. {5} is an element of A.
C. {4, 5} is an element of A.
D. {5} is a subset of A.
Answer» C. {4, 5} is an element of A.
discuss
3.

Which of these sets is finite?

A. {x | x is even}
B. ) {1, 2, 3,...}
C. {1, 2, 3,...,999,1000}
D. none
Answer» C. {1, 2, 3,...,999,1000}
discuss
4.

Which of these sets is not a null set?

A. A = {x | 6x = 24 and 3x = 1}
B. B = {x | x + 10 = 10}
C. C = {x | x is a man older than 200 years}
D. D = {x | x < x}
Answer» B. B = {x | x + 10 = 10}
discuss (1)
5.

. Let S={1, 2, 3}. How many subsets does S contain?

A. 3
B. 6
C. 8
D. 4
Answer» C. 8
discuss
6.

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

A. {{1},{2},{3}}
B. {1, 2, 3}
C. {{1, 2}, {2, 3}, {1, 3}}
D. {{1, 2, 3}}
Answer» B. {1, 2, 3}
discuss
7.

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

A. A U B ={v, w, x, y, z}
B. A U B = {v, w, y, z}
C. A U B = {v, w, x, y}
D. A U B ={x, w, x, y, z}
Answer» A. A U B ={v, w, x, y, z}
discuss
8.

which sets are equal ? 1.{r,s,t} 2.{s,s,t,r} 3.{t,r,t,s}

A. 1 and 2
B. 2 and 3
C. 1 and 3
D. all are equal
Answer» D. all are equal
discuss
9.

A U A=A according to …….law

A. Associative law
B. commutative law
C. Indempotent law
D. distributive law
Answer» C. Indempotent law
discuss
10.

In any application of the theory of sets, the members of all the sets belongs to …… set

A. union
B. intersection
C. universal
D. cardinal
Answer» C. universal
discuss
11.

let A={{a,b}} then aЄ A

A. TRUE
B. FALSE
C. Both
D. None
Answer» B. FALSE
discuss
12.

ФЄФ

A. TRUE
B. FALSE
C. Both
D. None
Answer» B. FALSE
discuss
13.

The sets {a,b,c} and {b,c,a} represnet the same sets.

A. TRUE
B. FALSE
C. Both
D. None
Answer» A. TRUE
discuss
14.

Cardinality of a set is number of element of the set.

A. TRUE
B. FALSE
C. Both
D. None
Answer» A. TRUE
discuss
15.

Multiset is an unordered collection of elemnts where an element can occur a a member more than once

A. TRUE
B. FALSE
C. Both
D. None
Answer» A. TRUE
discuss
16.

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

A. countably infinite
B. uncountably finite
C. countably finite
D. uncountably infinite
Answer» D. uncountably infinite
discuss
17.

Sum of first n positive integers is

A. n(n+1)
B. n
C. n(n+1)0.5
D. n(n+2)
Answer» C. n(n+1)0.5
discuss
18.

Let P(S) denote the power set of set S. which of the is always true

A. P(P(s))=p(s)
B. P(S)∩ S= P(S)
C. P(S)∩P(P(S)) ={Ф}
D. None
Answer» C. P(S)∩P(P(S)) ={Ф}
discuss
19.

{3}Є{1,3,5}

A. TRUE
B. FALSE
C. Both
D. None
Answer» B. FALSE
discuss
20.

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?

A. 100
B. 25
C. 56
D. 20
Answer» C. 56
discuss (1)
21.

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

A. Multiset
B. ordered set
C. set
D. None
Answer» A. Multiset
discuss
22.

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?

A. 9
B. 8
C. 7
D. 6
Answer» D. 6
discuss
23.

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

A. { 9, 10 }
B. {1, 2, 3 }
C. {1, 2, 3 9 }
D. {1, 2, 3 9 10 }
Answer» D. {1, 2, 3 9 10 }
discuss
24.

If U = {1, 2, 3, . . . 20 } and S = set of prime numbers , then S =

A. { 3, 5, 7, 11, 13, 17 }
B. { 2, 3, 5, 7, 11, 13, 17, 19 }
C. {1, 3, 5, 7, 9, 11, 13, 15, 17, 19 }
D. {1, 2, 3, 5, 7, 11, 13, 17 }
Answer» B. { 2, 3, 5, 7, 11, 13, 17, 19 }
discuss
25.

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

A. If n is not divisible by 30 then n is divisible by 2 or divisible by 3 or divisible by 5
B. If n is not divisible by 30 then n is not divisible by 2 or not divisible by 3 or not divisible by 5
C. If n is divisible by 2 and divisible by 3 and divisible by 5 then n is divisible by 30.
D. If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30
Answer» D. If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30
discuss
26.

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

A. If you lose the game then you don’t know the rules or you are overconfident.
B. A sufficient condition that you win the game is that you know the rules or you are not over confident
C. If you don’t know the rules or are overconfident you lose the game.
D. If you know the rules and are overconfiden t then you win the game.
Answer» A. If you lose the game then you don’t know the rules or you are overconfident.
discuss
27.

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

A. If T is a right triangle then a2 + b2 = c2.
B. If a2 + b2 = c2 then T is a right triangle.
C. If a2 + b2 6= c2 then T is not a right triangle.
D. T is a right triangle only if a2 + b2 = c2.
Answer» B. If a2 + b2 = c2 then T is a right triangle.
discuss
28.

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

A. I drink milk only if I do not eat a mango
B. If I don’t eat a mango then I drink milk
C. If I do not drink milk then I eat mango
D. None
Answer» B. If I don’t eat a mango then I drink milk
discuss
29.

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

A. If it is sultry then it is hot
B. It is sultry only if it is hot
C. It is sultry and it is not hot
D. None
Answer» D. None
discuss
30.

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

A. If a rectangle is not a a rhombus it is not a square
B. If a rhombus is not rectangle it is not a square
C. If a quadrilateral is neither a rectangle nor a rhombus then it is not a square
D. None
Answer» C. If a quadrilateral is neither a rectangle nor a rhombus then it is not a square
discuss
31.

For a conditional statement p===>q, which of the following is incorrect.

A. Converse of the inverse is its contrapositive
B. contrapositive of the converse is its inverse
C. Inverse of the contrapositiv e is its converse
D. None
Answer» D. None
discuss
32.

Which of the following is equivalent to p==>q

A. ~xp٨q
B. pV~q
C. ~xpVq
D. pV~q
Answer» C. ~xpVq
discuss
33.

Equivalent inverse of p==>q is

A. ~xp٨q
B. pV~q
C. ~xpVq
D. pV~q
Answer» D. pV~q
discuss
34.

Converse of p==>q is

A. ~xp٨q
B. pV~q
C. ~xpVq
D. pV~q
Answer» D. pV~q
discuss
35.

the truth table for exclusive disjunction will be

A. tautology
B. Contradiction
C. Logical equivalent
D. p or q but not both
Answer» D. p or q but not both
discuss
36.

The conditional statement p→q and its contrapositive are….

A. Converse
B. Inverse
C. Logically equivalent
D. None
Answer» C. Logically equivalent
discuss
37.

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

A. 3
B. 6
C. 9
D. None
Answer» B. 6
discuss
38.

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

A. 6
B. 9
C. 18
D. None
Answer» B. 9
discuss
39.

if P∩ Q= Ф then P U Q' is

A. P
B. U-P
C. U-Q
D. Ф
Answer» C. U-Q
discuss
40.

The number of proper subset of {1,2,3,4} is

A. 16
B. 15
C. 10
D. 12
Answer» A. 16
discuss
41.

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

A. A
B. B
C. C
D. D
Answer» C. C
discuss
42.

Which among the statements given in Q.70 is contradiction

A. A
B. B
C. C
D. D
Answer» D. D
discuss
43.

Consider 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. A
B. B
C. C
D. D
Answer» D. D
discuss
44.

consider 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. A
B. B
C. C
D. D
Answer» D. D
discuss
45.

In above question which statement is contradiction.

A. A
B. B
C. C
D. D
Answer» A. A
discuss
46.

Which 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}

A. 1 and 2 are equal
B. 2 and 3 are equal
C. 3 and 4 are equal
D. All are equal.
Answer» D. All are equal.
discuss
47.

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

A. 2 and 3 is true
B. 1 and 3 is true
C. 3 is true
D. None
Answer» A. 2 and 3 is true
discuss
48.

Determine 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.

A. Valid
B. Not valid
C. Both a and b
D. None
Answer» A. Valid
discuss
49.

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

A. 30
B. 52
C. 40
D. 68
Answer» B. 52
discuss
50.

In above Q.78 Find the number of people who read exactly 1 magzine.

A. 30
B. 52
C. 40
D. 68
Answer» A. 30
discuss
51.

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?

A. 73
B. 72
C. 50
D. 54
Answer» A. 73
discuss (1)
52.

In above q.80 how many students exactly know 2 languages?

A. 52
B. 54
C. 60
D. 25
Answer» B. 54
discuss
53.

In q. 80 how many students know exactly 1 language?

A. 54
B. 12
C. 7
D. 8
Answer» C. 7
discuss
54.

A preposition is a statement that is either ture or false

A. TRUE
B. FALSE
C. none
D. both a and b
Answer» A. TRUE
discuss
55.

A prepostition that is true under all circumstances is referred to as a ….

A. Tautology
B. Contradiction
C. Negation
D. Sentence
Answer» A. Tautology
discuss
56.

A prepostition that is false under all circumstances is referred to as a ….

A. Tautology
B. Contradiction
C. Negation
D. Sentence
Answer» B. Contradiction
discuss
57.

p→q is logically equivalent to ~p V q according to…

A. Identity law
B. Implication law
C. associative law
D. Absoption law
Answer» B. Implication law
discuss
58.

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

A. Disjunctive normal form
B. Conjunctive normal form
C. Normal form
D. None
Answer» B. Conjunctive normal form
discuss
59.

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

A. Disjunctive normal form
B. Conjunctive normal form
C. Normal form
D. None
Answer» A. Disjunctive normal form
discuss
60.

An assertion that contains one or more variable is called a….

A. CNF
B. DNF
C. Predicates
D. Quantifiers
Answer» C. Predicates
discuss
61.

Determine 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.

A. Valid
B. Invalid
C. Both a and b
D. none
Answer» B. Invalid
discuss
62.

p V ~(p٨q) is….

A. Contradiction
B. Tautology
C. predicate
D. None
Answer» B. Tautology
discuss
63.

Determine 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.

A. Valid
B. Invalid
C. Both a and b
D. none
Answer» A. Valid
discuss
64.

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

A. TRUE
B. FALSE
C. both a and b
D. none
Answer» B. FALSE
discuss
65.

De Morgan's laaws are two examplesof rules of inference

A. TRUE
B. FALSE
C. both a and b
D. none
Answer» B. FALSE
discuss
66.

In 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?

A. 250
B. 550
C. 120
D. 140
Answer» B. 550
discuss
67.

dual of (p V q)٨ r is..

A. p Vq
B. (p٨q) Vr
C. p ٨r
D. (p Vq) Vr
Answer» B. (p٨q) Vr
discuss
68.

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?

A. 10
B. 12
C. 35
D. 9
Answer» B. 12
discuss
69.

In above q.97 How many know exactly 2 languages?

A. 54
B. 16
C. 10
D. 35
Answer» A. 54
discuss
70.

In q.97. How many know exactly 1 language?

A. 6
B. 16
C. 7
D. 10
Answer» C. 7
discuss
71.

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?

A. 22
B. 45
C. 42
D. 14
Answer» C. 42
discuss
72.

[~ q ^ (p→q)]→~ p is,

A. Satisfiable
B. tautology
C. unsatisfiable
D. contradiction
Answer» B. tautology
discuss
73.

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

A. It is got and it is humid
B. It is hot and it is not humid
C. it is not hot and it is humid
D. none
Answer» B. It is hot and it is not humid
discuss
74.

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.

A. 10
B. 9
C. 8
D. 7
Answer» C. 8
discuss
75.

The statement ( p^q) → p is a

A. absurdity
B. contadiction
C. tautology
D. none
Answer» C. tautology
discuss
76.

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

A. p ^ q
B. ~ (~ p ^q)
C. p^ ~ q
D. ~ p ^q
Answer» B. ~ (~ p ^q)
discuss
77.

Let P(S) denotes the powerset of set S. Which of the following is always true?

A. P(P(S)) = P(S)
B. P(S) IS = P(S)
C. P(S) I P(P(S)) = {ø}
D. S € P(S)
Answer» D. S € P(S)
discuss
78.

Which of the following proposition is a tautology?

A. (p v q)→p
B. p v (q→p)
C. p v (p→q)
D. p→(p→q)
Answer» C. p v (p→q)
discuss
79.

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

A. 4 is odd and -5 is not negative
B. 4 is even or -5 is not negative
C. 4 is odd or -5 is not negative
D. 4 is even and -5 is not negative
Answer» A. 4 is odd and -5 is not negative
discuss
80.

Which one is the contrapositive of q → p ?

A. p → q
B. ~p →~q
C. ~q→~p
D. None
Answer» B. ~p →~q
discuss
81.

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

A. Invalid
B. Valid
C. Both a and b
D. None
Answer» B. Valid
discuss
82.

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

A. TRUE
B. FALSE
C. Both a and b
D. None
Answer» A. TRUE
discuss
83.

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

A. TRUE
B. FALSE
C. Both a and b
D. None
Answer» A. TRUE
discuss
84.

“If the sky is cloudy then it will rain and it will not rain”

A. absurdity
B. contadiction
C. tautology
D. none
Answer» C. tautology
discuss
85.

Represent 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”.

A. ∃x bird(x) V fly(x)
B. ∃x bird(x) ^ ~ fly(x)
C. ∃x bird(x) ^ fly(x)
D. None
Answer» B. ∃x bird(x) ^ ~ fly(x)
discuss
86.

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

A. ∀ (man(x)→ ~giant(x))
B. ∀ man(x)→ giant(x)
C. ∀ (man(x)→ giant(x))
D. None
Answer» C. ∀ (man(x)→ giant(x))
discuss
87.

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

A. ∃x man(x) ^ giant(x)
B. ∃x man(x) ^ ~ giant(x)
C. ∃x man(x) V ~ giant(x)
D. None
Answer» B. ∃x man(x) ^ ~ giant(x)
discuss
88.

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

A. ∃x [student(x) ^ likes(x, mathematics) ^~ likes(x, history)]Q.
B. ∃x [student(x) ^Vlikes(x, mathematics) V~ likes(x, history)]Q.
C. ∃x [student(x) ^ ~likes(x, mathematics) ^likes(x, history)]Q.
D. None
Answer» A. ∃x [student(x) ^ likes(x, mathematics) ^~ likes(x, history)]Q.
discuss
89.

AUB = (A− B)U(B−A)U(AпB).

A. FALSE
B. TRUE
C. Both a and b
D. None
Answer» B. TRUE
discuss
90.

[(PVQ)^(P→R)^(Q→S)] → (SVR). Is a….

A. absurdity
B. contadiction
C. tautology
D. none
Answer» C. tautology
discuss
91.

~(x vy) = ~x ^ ~y

A. FALSE
B. TRUE
C. Both a and b
D. None
Answer» B. TRUE
discuss
92.

(x ^ y)’ = x’ V y’

A. FALSE
B. TRUE
C. Both a and b
D. None
Answer» B. TRUE
discuss
93.

Test 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”

A. Invalid
B. Valid
C. Both a and b
D. None
Answer» B. Valid
discuss
94.

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?

A. 78
B. 98
C. 23
D. 43
Answer» D. 43
discuss
95.

In above Q.123 (ii) How many play only foot ball?

A. 10
B. 8
C. 9
D. 4
Answer» A. 10
discuss
96.

(p ↔ q) ↔ r = p ↔ (q ↔ r)

A. absurdity
B. contadiction
C. tautology
D. none
Answer» C. tautology
discuss
97.

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

A. If Jack eat and broccoli then he did ate fat.
B. If Jack did not eat broccoli then he did ate fat.
C. If Jack did not eat broccoli or he did ate fat.
D. If Jack did not eat broccoli then he did not ate fat.
Answer» B. If Jack did not eat broccoli then he did ate fat.
discuss
98.

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

A. The weather is not bad or I will go to work.
B. The weather is good or I will go to work.
C. The weather is not bad or I will not go to work.
D. None
Answer» A. The weather is not bad or I will go to work.
discuss
99.

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

A. Mary did loss her lamb and the wolf eat the lamb.
B. Mary did loss her lamb and the wolf did not eat the lamb.
C. Mary did not loss her lamb and the wolf did not eat the lamb.
D. None
Answer» C. Mary did not loss her lamb and the wolf did not eat the lamb.
discuss
100.

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

A. I will not win the game and I will enter the contest.
B. I will win the game and I will enter the contest.
C. I will win the game and I will not enter the contest.
D. None
Answer» B. I will win the game and I will enter the contest.
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