Discrete Structures and Optimization | Multiple Choice Questions

by Yogi P - December 24, 2024

Discrete Structures and Optimization

MCQs on Topic: Mathematical Logic

Which one of the following is a tautology?

(A) 𝑃∨¬ 𝑃
(B) 𝑃∧¬ 𝑃
(C) P∨Q
(D) ¬𝑃∨¬𝑄

Answer: A

What is truth value of P⟹Q if 𝑃 is false and Q is true?

(A) True
(B) False
(C) Cannot be determined
(D) Both True and False

Answer: A

Which Normal Form (NF) is obtained by using distributive laws in propositional logic?

(A) Disjunctive Normal Form (DNF)
(B) Conjunctive Normal Form (CNF)
(C) Both DNF and CNF
(D) None of these

Answer: C

In the predicate logic, the formula ∀x(P(x)⟹Q(x)) is equivalent to:

(A) ∃x(P(x)∧¬Q(x))
(B) ∀x(¬P(x)∨Q(x))
(C) ¬∀x(P(x)∧Q(x))
(D) ∀x(¬P(x)∨¬Q(x))

Answer: B

A formula that is true under every interpretation is called:

(A) Satisfiable
(B) Contradiction
(C) Contingent
(D) Tautology

Answer: D

Which one of the following logical equivalences is correct?

(A) P∨(P∧Q)=P
(B) P∧(P∨Q)=Q
(C) P∨(P∨Q)=Q
(D) P∧(P∨Q)=P∨Q

Answer: A

The expression (P∨Q)∧(¬P∨R) is equivalent to:

(A) P∧Q∨R
(B) (P∧R)∨(Q∧R)
(C) (P∨R)∧(Q∨R)
(D) (P∨Q)∨R

Answer: C

The formula P∧¬P is:

(A) A tautology
(B) A contradiction
(C) Satisfiable
(D) None of these

Answer: B

Which of the following statements are logically equivalent?

(i) P∨¬Q
(ii) ¬P∧Q
(iii) ¬(P⟹Q)

(A) Only (i) and (ii)
(B) Only (ii) and (iii)
(C) Only (i) and (iii)
(D) All of these

Answer: C

Discrete Structures and Optimization

Topic: Set Theory set of 10 MCQs:

Which of the following is a power set of {a,b}?

(A) {{a,b}}
(B) {{},{a},{b},{a,b}}
(C) {{},{a},{a,b}}
(D)  {{a},{b},{a,b}}

Answer: B

If A={1,2,3} and B={3,4,5}, what is  A∩B?

(A) {1,2}
(B) {3}
(C) {3,4}
(D) {1,2,3}

Answer: B

The universal set U contains {1,2,3,4,5,6,7,8,9}. If A={2,4,6,8}, what is Ac?

(A) {1,3,5,7,9}
(B) {2,4,6,8}
(C) {1,3,5,7}
(D) {2,4,5,6}

Answer: A

If ∣A∣=4, ∣B∣=3, and ∣A∩B∣=2, what is ∣A∪B∣?

(A) 6
(B) 5
(C) 7
(D) 8

Answer: B

Let A={1,2,3}, B={3,4,5}. The symmetric difference AΔB is:

(A) {1,2,3,4,5}
(B) {1,2,4,5}
(C) {3}
(D) {1,2,4}

Answer: B

Which one of the following represents the Cartesian product A×B for A={1,2} and B={x,y}?

(A) {{1,x},{2,y}}
(B) {{1,2},{x,y}}
(C)  {(1,x),(1,y),(2,x),(2,y)}
(D)  {(x,1),(y,2)}

Answer: C

If A={x∣x is a prime number less than 10}, what is A?

(A) {2,3,5,7}
(B) {1,2,3,5,7}
(C) {1,2,3,7}
(D) {3,5,7,11}

Answer: A

If A⊆B, which of the following is always true?

(A) A∪B=B
(B) A∩B=A
(C) Both A and B
(D) None of these

Answer: C

The number of subsets of set A containing n elements is:

(A) 2n
(B) n²
(C) 2ⁿ
(D) n!

Answer: C

If A={1,2} and B={2,3}, which of the following is true?

(A) A∩B={2}
(B) A∪B={1,2,3}
(C) A−B={1}
(D) All of these

Answer: D