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## Discrete Mathematics MCQ Set 1

1. For some number b, (1b)-n is equal to:
a) -bn
b) nb
c) bn
d) None of the mentioned

Answer: c [Reason:] b-1 reciprocal of b.

2. If ab = 1, where a and b are real numbers then:
a) a = b-1
b) b = a
c) a = b = 2
d) None of the mentioned

Answer: a [Reason:] This means that a is inverse of b or b is inverse of a.

3. If a is a real number than a0 is defined as:
a) 0
b) a
c) 1
d) None of the mentioned

Answer: a [Reason:] Any number to the power zero is one.

4. For some number a, b and c, ca x cb is equal to :
a) ca-b
b) ca+b
c) c
d) None of the mentioned

Answer: b [Reason:] If base are same then exponenents powers are added.

5. For some number a, b and c, ca/cb is equal to :
a) ca-b
b) ca+b
c) c
d) None of the mentioned

Answer: a [Reason:] If base are same then exponenents powers are added, 1/cb = c-b.

6. State whether the given statement is true or false
Exponentiation is commutative.
a) True
b) False

Answer: b [Reason:] Ab is not equal to bA, exponentiation is not commutative.

7. State whether the given statement is true or false
Exponentiation is associative.
a) True
b) False

Answer: b [Reason:] Exponentiation is not associative.

8. If 2a-b = 1 then the value of a-b is equal to :
a) 1
b) 0
c) 2
d) none of the mentioned

Answer: b [Reason:] 1 = 20, so a-b = 0.

9. For some number a, b and c, ac x bc is equal to :
a) (ab)c
b) (ac)b
c) (cb)a
d) None of the mentioned.

Answer: a [Reason:] If power are same then bases are multiplied.

10. If 0a is not equal to zero then which of the values a cannot take :
a) 1
b) 2
c) -1
d) 0

Answer: d [Reason:] a0 = 1, for any real number.

## Discrete Mathematics MCQ Set 2

1. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then number of elements in A U B is
a) 4
b) 5
c) 6
d) 7

Answer: a [Reason:] AUB is {1, 2, 3, 4}.

2. Let the set A is {1, 2, 3} and B is { 2, 3, 4}. Then number of elements in A ∩ B is
a) 1
b) 2
c) 3
d) 4

Answer: b [Reason:] A ∩ B is {2, 3}.

3. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then the set A – B is
a) {1, -4}
b) {1, 2, 3}
c) {1}
d) {2, 3}

Answer: c [Reason:] In A – B the common elements get cancelled.

4. In which of the following sets A- B is equal to B – A
a) A= {1, 2, 3}, B ={2, 3, 4}
b) A= {1, 2, 3}, B ={1, 2, 3, 4}
c) A={1, 2, 3}, B ={2, 3, 1}
d) A={1, 2, 3, 4, 5, 6}, B ={2, 3, 4, 5, 1}

Answer: c [Reason:] A- B= B-A = Empty set.

5. Let A be set of all prime numbers, B be the set of all even prime numbers, C be the set of all odd prime numbers, then which of the following is true?
a) A ≡ B U C
b) B is a singleton set.
c) A ≡ C U {2}
d) All of the mentioned

Answer: d [Reason:] 2 is the only even prime number.

6. If A has 4 elements B has 8 elements then the minimum and maximum number of elements in A U B are respectively
a) 4, 8
b) 8, 12
c) 4, 12
d) None of the mentioned

Answer: b [Reason:] Minimum would be when 4 elements are same as in 8, maximum would be when all are distinct.

7. If A is {{Φ}, {Φ, {Φ}}, then the power set of A has how many element?
a) 2
b) 4
c) 6
d) 8

Answer: b [Reason:] The set A has got 2 elements so n(P(A))=4.

8. Two sets A and B contains a and b elements respectively .If power set of A contains 16 more elements than that of B, value of ‘b’ and ‘a’ are respectively
a) 4, 5
b) 6, 7
c) 2, 3
d) None of the mentioned

Answer: a [Reason:] 32-16=16, hence a=5, b=4.

9. Let A be {1, 2, 3, 4}, U be set of all natural numbers, then U-A’(complement of A) is given by set.
a) {1, 2, 3, 4, 5, 6, ….}
b) {5, 6, 7, 8, 9, ……}
c) {1, 2, 3, 4}
d) All of the mentioned

Answer: c [Reason:] U – A’ ≡ A.

10. Which sets are not empty?
a) {x: x is a even prime greater than 3}
b) {x : x is a multiple of 2 and is odd}
c) {x: x is an even number and x+3 is even}
d) { x: x is a prime number less than 5 and is odd}

Answer: d [Reason:] Because the set is {3}.

## Discrete Mathematics MCQ Set 3

1. Let P (x) denote the statement “x >7.” Which of these have truth value true?
a) P (0)
b) P (4)
c) P (6)
d) P (9)

Answer: d [Reason:] Put x=9, 9>7 which is true.

2. Let Q(x) be the statement “x < 5.” What is the truth value of the quantification ∀xQ(x), having domains as real numbers?
a) True
b) False

Answer: b [Reason:] Q(x) is not true for every real number x, because, for instance, Q(6) is false. That is, x = 6 is a counterexample for the statement ∀xQ(x). Thus is false.

3. Determine the truth value of ∀n(n + 1 > n) if the domain consists of all real numbers.
a) True
b) False

Answer: a [Reason:] There are no elements in the domain for which the statement is false.

4. Let P(x) denote the statement “x = x + 7.” What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?
a) True
b) False

Answer: b [Reason:] Because P(x) is false for every real number x, the existential quantification of Q(x), which is ∃xP(x), is false.

5. Let R (x) denote the statement “x > 2.” What is the truth value of the quantification ∃xR(x), having domain as real numbers?
a) True
b) False

Answer: a [Reason:] Because “x > 2” is sometimes true—for instance, when x = 3–the existential quantification of R(x), which is ∃xR(x), is true.

6. 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 [Reason:] For every person x, if comedian then x is funny.

7. 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 [Reason:] For some x, x is friend and funny.

8. ”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 [Reason:] Domain may be limited to your class or may be whole world both are good as it satisfies universal quantifier.

9. 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 [Reason:] There is no student who spends more than 2 hours in playing polo.

10. Determine the truth value of statement ∃n (4n = 3n) if the domain consists of all integers.
a) True
b) False

Answer: a [Reason:] For n=0, 4n=3n hence, it is true.

## Discrete Mathematics MCQ Set 4

1. Let P and Q be statements, then P<->Q is logically equivalent to
a) P<->~Q
b) ~P<->Q
c) ~P<->~Q
d) None of the mentioned

Answer: c [Reason:] Both of them have same truth table, Hence they are equal.

2. What is the negation of the statement A->(B v(or) C)?
a) A ∧ ~B ∧ ~C
b) A->B->C
c) ~A ∧ B v C
d) None of the mentioned

Answer: a [Reason:] A->P is logically equivalent to ~A v P.

3. The compound statement A-> (A->B) is false, then the truth values of A, B are respectively
a) T, T
b) F, T
c) T, F
d) F, F

Answer: c [Reason:] For implications to be false hypothesis should be true and conclusion should be false.

4. The statement which is logically equivalent to A∧ (and) B is
a) A->B
b) ~A ∧ ~ B
c) A ∧ ~B
d) ~(A->~B)

Answer: d [Reason:] The truth table of both statements are same.

5. Let P: We give a nice overall squad performance, Q: We will win the match.
Then the symbolic form of “We will win the match if and only if we give a nice overall squad performance. “is
a) P v Q
b) Q ∧ P
c) Q<->P
d) ~P v Q

Answer: c [Reason:] If and only if statements are bi-conditionals.

6. Let P, Q, R be true, false true , respectively, which of the following is true
a) P∧Q∧R
b) P∧~Q∧~R
c) Q->(P∧R)
d) P->(Q∧R)

Answer: c [Reason:] Hypothesis is false, hence statement is true.

7. “Match will be played only if it is not a humid day.” The negation of this statement is
a) Match will be played but it is a humid day
b) Match will be played or it is a humid day
c) All of the mentioned statement are correct
d) None of the mentioned.

Answer: a [Reason:] Negation of P->Q is P∧~Q.

8. Consider the following statements
A: Raju should exercise.
B: Raju is not a decent table tennis player.
C: Raju wants to play good table tennis.
The symbolic form of “Raju is not a decent table tennis player and if he wants to play good table tennis then he should exercise.” is
a) A->B->C
b) B∧(C->A)
c) C->B∧A
d) B<->A∧C

Answer: b [Reason:] For conditionals statement (if then), implications are used.

9. The statement (~P<->Q)∧~Q is false when
a) P:True Q: False
b) P:True Q:True
c) P:False Q:True
d) P :False Q:False

Answer: b [Reason:] For a bi-conditional to be true both inputs should be same.

10. Let P, Q, R be true, false, false, respectively, which of the following is true
a) P∧(Q∧~R)
b) (P->Q)∧~R
c) Q<->(P∧R)
d) P<->(QvR)

Answer: c [Reason:] For a bi-conditional to be true both inputs should be same.

## Discrete Mathematics MCQ Set 5

1. Let A1, A2, be two AM’s and G1, G2 be two GM’s between a and b,then (A1 + A2) / G1G2 is equal to :
a) (a+b) / 2ab
b) 2ab/(a+b)
c) (a+b)/(ab)
d) None of the mentioned

Answer: c [Reason:] A1 + A2 = a + b, G1G2 = ab.

2. The series a,(a+b)/2, b is in
a) AP
b) GP
c) HP
d) None of the mentioned

Answer: a [Reason:] (a+b)/2 is AM between a, b. Hence series is in AP.

3. The series a, (ab)1/2, b is in
a) AP
b) GP
c) HP
d) None of the mentioned

Answer: b [Reason:] (ab)1/2 is GM between a, b. Hence series is in GP.

4. If A and G be the A.M and G.M between two positive number then the numbers are A + (A2 – G2)1/2, A – (A2 – G2)1/2 . The given statement is
a) True
b) False

Answer: a [Reason:] The equation having its roots as given equation is x2 – 2Ax + G2 = 0 which implies x = A + (A2 – G2)1/2, A – (A2 – G2)1/2.

5. If one geometric mean G and two airthmetic mean A1, A2 are inserted between two numbers,then (2A1 – A2) (2A2 – A1) is equal to
a) 2G
b) G
c) G2
4) None of the mentioned.

Answer: c [Reason:] Let a and b be two numbers then, G = (ab)1/2, A1 = (2a+b)/3, A2 = (a+2b)/3, (2A1 – A2) = a, (2A2 – A1) = b, (2A1 – A2)(2A2 – A1) = G2.

6. State whether the given statement is true or false
AM ≤ GM.
a) True
b) False

Answer: b [Reason:] Airthmetic Mean is always greater or equal to geometric mean.

7. If between two numbers which are root of given equation
x2 – 18x + 16 = 0, a GM is inserted then the value of that GM is.
a) 4
b) 5
c) 6
d) 16

Answer: a [Reason:] x2 – 2Ax + G2 = 0, here G2 = 16 and therefore G = 4.

8. If a1, a2, a3 are in airthemetic as well as geometric progression then which of the following is/are correct ?
a) 2a2 = a1 + a3
b) a2 = (a1a3)1/2
c) a2 – a1 = a3 -a2
d) All of the mentioned are correct.

Answer: d [Reason:] a2 is AM, GM between a1, a3, also the series is in AP so common difference should be same.

9. If a1, a2, a3 are in GP then 1/a1, 1/a2, 1/a3 are in :
a) AP
b) GP
c) HP
d) None of the mentioned.