# 440+ Discrete Mathematics Solved MCQs

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1.

A. relation
B. function
C. set
D. proposition
2.

A. one
B. two
C. zero
D. three
3.

## 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. {1, 3, 5, 7, 9}
4.

A. 10
B. 5
C. 3
D. 20
5.

## 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. a = {1, 2, 3} and b = {2, 1, 3}
6.

A. infinite
B. finite
C. subset
D. empty
7.

A. 8
B. 6
C. 7
D. 9
8.

## 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. {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
9.

## The union of the sets {1, 2, 5} and {1, 2, 6} is the set _______________.

A. {1, 2, 6, 1}
B. {1, 2, 5, 6}
C. {1, 2, 1, 2}
D. {1, 5, 6, 3}
Answer» B. {1, 2, 5, 6}
10.

## The intersection of the sets {1, 2, 5} and {1, 2, 6} is the set ___________.

A. {1, 2}
B. {5, 6}
C. {2, 5}
D. {1, 6}
Answer» A. {1, 2}
11.

## Two sets are called disjoint if there _____________ is the empty set.

A. union complement
B. difference
C. intersection
D. complement
12.

## Which of the following two sets are disjoint?

A. {1, 3, 5} and {1, 3, 6}
B. {1, 2, 3} and {1, 2, 3}
C. {1, 3, 5} and {2, 3, 4}
D. {1, 3, 5} and {2, 4, 6}
Answer» D. {1, 3, 5} and {2, 4, 6}
13.

A. {1}
B. {5}
C. {3}
D. {2}
14.

A. a – b
B. u – a
C. a – u
D. b – a
Answer» B. u – a
15.

A. 1010100000
B. 1010101101
C. 1111111100
D. 1111101010
16.

A. A
B. null
C. U
D. B
17.

## If A = {a,b,{a,c}, ∅}, then A - {a,c} is

A. {a, b, ∅}
B. {b, {a, c}, ∅}
C. {c, {b, c}}
D. {b, {a, c}, ∅}
Answer» A. {a, b, ∅}
18.

## The set (A - B) – C is equal to the set

A. (a – b) ∩ c
B. (a∪ b) – c
C. (a – b) ∪ c
D. (a ∪ b) – c
Answer» D. (a ∪ b) – c
19.

A. 100
B. 120
C. 130
D. 140
20.

## Using Induction Principle if 13 = 1, 23 = 3 + 5, 33 = 7 + 9 + 11, then

A. 43= 15 + 17 + 19 + 21
B. 43= 11 + 13 + 15 + 17 + 19
C. 43 = 13 + 15 + 17 + 19
D. 43 = 13 + 15 + 17 + 19 + 21
Answer» C. 43 = 13 + 15 + 17 + 19
21.

## By mathematical Induction 2n> n3

A. for n ≥ 1
B. for n ≥ 4
C. for n ≥ 5
D. for n ≥ 10
Answer» D. for n ≥ 10
22.

## The symmetric difference A ⊕ B is the set

A. a – a ∩ b
B. (a∪ b) – (a∩ b)
C. (a – b) ∩ (b – a)
D. a ∪ (b – a)
Answer» B. (a∪ b) – (a∩ b)
23.

A. a ⊕ b
B. a ∪ b
C. a – b
D. a ∩ b
Answer» A. a ⊕ b
24.

## 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
Answer» C. a  b
25.

A. n
B. 2n - 1
C. n2
D. 2n
26.

## 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. {1, 3, 5, 7, 9}
27.

A. infinite
B. finite
C. subset
D. empty
28.

## If p ˄ q is T, then

A. p is t, q is t
B. p is f, q is t
C. p is f, q is f
D. p is t, q is f
Answer» B. p is f, q is t
29.

## If p →q is F, then

A. p is t, q is t
B. p is f, q is t
C. p is f, q is f
D. p is t, q is f
Answer» D. p is t, q is f
30.

## The statement from ∼ (p ˄ q) is logically equivalent to

A. ∼ p ˅ ∼ q
B. ∼ p ˅ qc
C. p ˅ ∼ q
D. ∼ p ˄∼ q
Answer» A. ∼ p ˅ ∼ q
31.

A. p
B. tautology
D. none of these
32.

A. ∼q → ∼p
B. ∼ p → ∼ q
C. ∼ p → q
D. q → p
Answer» D. q → p
33.

## Let p: Mohan is rich, q : Mohan is happy, then the statement: Mohan is rich, but Mohan is not happy in symbolic form is

A. p ˄ q
B. ∼ p˄ q
C. p ˅ q
D. p ˄ ∼ q
Answer» D. p ˄ ∼ q
34.

A. p → q
B. p ˄ q
C. q → p
D. p ˄ q
Answer» A. p → q
35.

## Let p denote the statement: “Gopal is tall”, q: “Gopal is handsome”. Then the negation of the statement Gopal is tall, but not handsome,in symbolic form is:

A. ∼ p ˄q
B. ∼ p ˅ q
C. ∼ p ˅∼q
D. ∼ p ˄∼q
Answer» B. ∼ p ˅ q
36.

## If p ˄ (p → q) is T, then

A. p is t
B. p is f, q is t
C. p is t, q is t
D. p is f, q is f
Answer» C. p is t, q is t
37.

## If (∼ (p ˅ q)) → q is F, then

A. p is t, q is f
B. p is f, q is t
C. p is t, q is t
D. p is f, q is
Answer» B. p is f, q is t
38.

## If (∼ p → r) ˄ (p ↔ q) is T and r is F, then truth values of p and q are:

A. p is t, q is t
B. p is t, q is f
C. p is f, q is f
D. p is f, q is t
Answer» A. p is t, q is t
39.

## If ((p → q ) → q) → p is F, then

A. p is t, q is t
B. p is t, q is f
C. p is f, q is t
D. p is f, q is f
Answer» C. p is f, q is t
40.

## (p ˄ (p → q )) → q is logically equivalent to

A. p ˅ q
B. (p ˄ q) ˅ (~ p˄ ~q)
C. tautology
D. (~ p ˅ q) ˄ (p ˅ q)
41.

## If (p ˅ q) ˄ (~ p˅ ~q) is F, then

A. p is t, q is t, or q is f
B. p is f, q is t
C. p is t, q is f
D. p and q must have same truth values
Answer» D. p and q must have same truth values
42.

## Let p denote the statement: “I finish my homework before dinner”, q: “It rains” and r: “I will go for a walk”, the representative of the following statement: if I finish my homework before dinner and it does not rain, then I will go for walk is

A. p ˄ ~q ˄ r
B. (p ˄ ~q )→ r
C. p →(~q˄ r)
D. (p →~q)→ r)
Answer» B. (p ˄ ~q )→ r
43.

## The contrapositive of p →q is

A. ~ q → ~ p
B. ~ p → ~ qc
C. ~ p → q
D. ~ q → p
Answer» A. ~ q → ~ p
44.

## Which of the following is declarative statement?

A. it’s right
B. three is divisible by 3.
C. two may not be an even integer
D. i love you
Answer» B. three is divisible by 3.
45.

## Which of the proposition is p ^ (~p v q) is

A. tautulogy
C. logically equivalent to p ^ q
D. all of above
Answer» C. logically equivalent to p ^ q
46.

## 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}
Answer» D. range of r = {5}
47.

## 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)}
C. {(3, 3), (4, 3), (5, 4), (3, 4)}
D. none of the above
Answer» D. none of the above
48.

## If R = {x, y) : x, y Î 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
Answer» C. {– 2, – 1, 0, 1, 2}
49.

## 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
Answer» D. not transitive
50.

## 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
51.

A. reflexive
B. symmetric
C. transitive
D. equivalence
52.

## Let a relation R in the set R of real numbers be defined as (a, b) Î R if and only if 1 + ab > 0 for all a, bÎR. The relation R is

A. reflexive and symmetric
B. symmetric and transitive
C. only transitive
D. an equivalence relation
Answer» A. reflexive and symmetric
53.

## If R be relation ‘<‘ from A = {1, 2, 3, 4} to B = {1, 3, 5} ie, (a, b) Î R iff a < b, then RoR– 1 is

A. {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}
B. {(3, 1), (5, 1), (3, 2), (5, 2), (5, 3), (5, 4)}
C. {(3, 3), (3, 5), (5, 3), (5, 5)}
D. { (3, 3), (3, 4), (4, 5)}
Answer» C. {(3, 3), (3, 5), (5, 3), (5, 5)}
54.

## R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3. The relation R – 1 is

A. {(11, 8), (13, 10)}
B. {(8, 11), (10, 13)}
C. {(8, 11), (9, 12), (10, 13)}
D. none of the above
Answer» B. {(8, 11), (10, 13)}
55.

## R is a relation on N given by N = {(x, y): 4x + 3y = 20}. Which of the following belongs to R?

A. (– 4, 12)
B. (5, 0)
C. (3, 4)
D. (2, 4)
Answer» D. (2, 4)
56.

## The relation R defined on the set of natural numbers as {(a, b): a differs from b by 3} is given

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
Answer» B. { (4, 1), (5, 2), (6, 3), ….}
57.

A. 7
B. 9
C. 10
D. 12
58.

A. 4
B. 6
C. 8
D. 12
59.

## Let R = { ( 3, 3 ) ( 6, 6 ) ( ( 9, 9 ) ( 12, 12 ), ( 6, 12 ) ( 3, 9 ) ( 3, 12 ), ( 3, 6 ) } be a relation on the set A = { 3, 6, 9, 12 }. The relation is

A. reflexive and transitive
B. reflexive only
C. an equivalence relation
D. reflexive and symmetric only
Answer» A. reflexive and transitive
60.

A. (0,π/2)
B. (0,(-π)/2)
C. (π/2,(-π)/2)
D. ((-π)/2,π/2)
61.

## Let R be the set of real numbers. If f : R → R is a function defined by f ( x ) = x2 , then f is]

A. inject ve but not subjective
B. subjective but not injective
C. bijective
D. none of these
Answer» D. none of these
62.

## Which of the following statement is a proposition?

A. get me a glass of milkshake
B. god bless you!
C. what is the time now?
D. the only odd prime number is 2
Answer» D. the only odd prime number is 2
63.

A. 1
B. 3
C. 0
D. 2
64.

## Let P: I am in Bangalore.; Q: I love cricket.; then q -> p(q implies p) is?

A. if i love cricket then i am in bangalore
B. if i am in bangalore then i love cricket
C. i am not in bangalore
D. i love cricket
Answer» A. if i love cricket then i am in bangalore
65.

## Let P: If Sahil bowls, Saurabh hits a century.; Q: If Raju bowls, Sahil gets out on first ball. Now if P is true and Q is false then which of the following can be true?

A. raju bowled and sahil got out on first ball
B. raju did not bowled
C. sahil bowled and saurabh hits a century
D. sahil bowled and saurabh got out
Answer» C. sahil bowled and saurabh hits a century
66.

## Let P: I am in Delhi.; Q: Delhi is clean.; then q ^ p(q and p) is?

A. delhi is clean and i am in delhi
B. delhi is not clean or i am in delhi
C. i am in delhi and delhi is not clean
D. delhi is clean but i am in mumbai
Answer» A. delhi is clean and i am in delhi
67.

## Let P: This is a great website, Q: You should not come back here. Then ‘This is a great website and you should come back here.’ is best represented by?

A. ~p v ~q
B. p ∧ ~q
C. p v q
D. p ∧ q
Answer» B. p ∧ ~q
68.

## Let P: We should be honest., Q: We should be dedicated., R: We should be overconfident. Then ‘We should be honest or dedicated but not overconfident.’ is best represented by?

A. ~p v ~q v r
B. p ∧ ~q ∧ r
C. p v q ∧ r
D. p v q ∧ ~r
Answer» D. p v q ∧ ~r
69.

A. p ↔ q
B. p → q
C. ¬ (p ∨ q)
D. ¬p ∨ ¬q
Answer» A. p ↔ q
70.

## p → q is logically equivalent to

A. ¬p ∨ ¬q
B. p ∨ ¬q
C. ¬p ∨ q
D. ¬p ∧ q
Answer» C. ¬p ∨ q
71.

## p ∨ q is logically equivalent to

A. ¬q → ¬p
B. q → p
C. ¬p → ¬q
D. ¬p → q
Answer» D. ¬p → q
72.

A. q↔p
B. p↔¬q
C. ¬p↔¬q
D. ¬q↔¬p
73.

## Which of the following statement is correct?

A. p ∨ q ≡ q ∨ p
B. ¬(p ∧ q) ≡ ¬p ∨ ¬q
C. (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
D. all of mentioned
Answer» D. all of mentioned
74.

## p ↔ q is logically equivalent to

A. (p → q) → (q → p)
B. (p → q) ∨ (q → p)
C. (p → q) ∧ (q → p)
D. (p ∧ q) → (q ∧ p)
Answer» C. (p → q) ∧ (q → p)
75.

## (p → q) ∧ (p → r) is logically equivalent to

A. p → (q ∧ r)
B. p → (q ∨ r)
C. p ∧ (q ∨ r)
D. p ∨ (q ∧ r)
Answer» A. p → (q ∧ r)
76.

## (p → r) ∨ (q → r) is logically equivalent to

A. (p ∧ q) ∨ r
B. (p ∨ q) → r
C. (p ∧ q) → r
D. (p → q) → r
Answer» C. (p ∧ q) → r
77.

A. p (0)
B. p (4)
C. p (6)
D. p (9)
Answer» D. p (9)
78.

## The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.

A. ∃x(c(x) ∧ f (x))
B. ∀x(c(x) ∧ f (x))
C. ∃x(c(x) → f (x))
D. ∀x(c(x) → f (x))
Answer» D. ∀x(c(x) → f (x))
79.

## The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.

A. ∀x (f (x) → p (x))
B. ∀x (f (x) ∧ p (x))
C. ∃x (f (x) ∧ p (x))
D. ∃x (f (x) → p (x))
Answer» C. ∃x (f (x) ∧ p (x))
80.

## ”Everyone wants to learn cosmology.” This argument may be true for which domains?

A. all students in your cosmology class
B. all the cosmology learning students in the world
C. both of the mentioned
D. none of the mentioned
Answer» C. both of the mentioned
81.

## Let domain of m includes all students, P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.

A. a student is there who spends more than 2 hours in playing polo
B. there is a student who does not spend more than 2 hours in playing polo
C. all students spends more than 2 hours in playing polo
D. no student spends more than 2 hours in playing polo
Answer» D. no student spends more than 2 hours in playing polo
82.

## Translate ∀x∃y(x < y) in English, considering domain as a real number for both the variable.

A. for all real number x there exists a real number y such that x is less than y
B. for every real number y there exists a real number x such that x is less than y
C. for some real number x there exists a real number y such that x is less than y
D. for each and every real number x and y such that x is less than y
Answer» A. for all real number x there exists a real number y such that x is less than y
83.

## “The product of two negative real numbers is not negative.” Is given by?

A. ∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
B. ∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
C. ∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
D. ∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
Answer» D. ∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
84.

## Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”

A. ∀x l(x, joy)
B. ∀y l(joy,y)
C. ∃y∀x l(x, y)
D. ∃x ¬l(joy, x)
Answer» A. ∀x l(x, joy)
85.

## Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.

A. all students does not like south indian dishes.
B. amit does not like south indian people.
C. amit does not like south indian dishes.
D. amit does not like some dishes.
Answer» C. amit does not like south indian dishes.
86.

## Use quantifiers and predicates with more than one variable to express, “There is a pupil in this lecture who has taken at least one course in Discrete Maths.”

A. ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
B. ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all discrete maths lectures, and the domain for y consists of all pupil in this class
C. ∀x∀yp(x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
D. ∃x∀yp(x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
Answer» A. ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
87.

## Find a counterexample of ∀x∀y(xy > y), where the domain for all variables consists of all integers.

A. x = -1, y = 17
B. x = -2 y = 8
C. both x = -1, y = 17 and x = -2 y = 8
D. does not have any counter example
Answer» C. both x = -1, y = 17 and x = -2 y = 8
88.

## Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”

A. modus tollens
B. modus ponens
C. disjunctive syllogism
D. simplification
Answer» B. modus ponens
89.

## Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”

A. modus tollens
B. conjunction
C. hypothetical syllogism
D. simplification
Answer» A. modus tollens
90.

## Which rule of inference is used, ”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”

A. simplification
B. conjunction
D. disjunctive syllogism
91.

## What rules of inference are used in this argument? “All students in this science class has taken a course in physics” and “Marry is a student in this class” imply the conclusion “Marry has taken a course in physics.”

A. universal instantiation
B. universal generalization
C. existential instantiation
D. existential generalization
Answer» A. universal instantiation
92.

## What rules of inference are used in this argument? “It is either colder than Himalaya today or the pollution is harmful. It is hotter than Himalaya today. Therefore, the pollution is harmful.”

A. conjunction
B. modus ponens
C. disjunctive syllogism
D. hypothetical syllogism
Answer» C. disjunctive syllogism
93.

A. p ∨ r
B. p ∨ s
C. p ∨ q
D. q ∨ r
Answer» B. p ∨ s
94.

## What rules of inference are used in this argument? “Jay is an awesome student. Jay is also a good dancer. Therefore, Jay is an awesome student and a good dancer.”

A. conjunction
B. modus ponens
C. disjunctive syllogism
D. simplification
95.

## “Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that

A. parul is out for trip
B. raju is playing chess
C. parul is out for a trip and raju is playing chess
D. parul is out for a trip or raju is playing chess
Answer» D. parul is out for a trip or raju is playing chess
96.

## Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove

A. ∀np ((n) → q(n))
B. ∃ np ((n) → q(n))
C. ∀n~(p ((n)) → q(n))
D. ∀np ((n) → ~(q(n)))
Answer» A. ∀np ((n) → q(n))
97.

## Which of the following can only be used in disproving the statements?

A. direct proof
B. contrapositive proofs
C. counter example
D. mathematical induction
Answer» C. counter example
98.

## When to proof P→Q true, we proof P false, that type of proof is known as

A. direct proof
B. contrapositive proofs
C. vacuous proof
D. mathematical induction
Answer» C. vacuous proof
99.

## In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?

A. direct proof
B. proof by contradiction
C. vacuous proof
D. mathematical induction
100.

## A proof covering all the possible cases, such type of proofs are known as

A. direct proof
B. proof by contradiction
C. vacuous proof
D. exhaustive proof
Answer» D. exhaustive proof