# Discrete Mathematics Questions and Answers – Logics – Logical Equivalences

This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Logics – Logical Equivalences”.

1. The compound propositions p and q are called logically equivalent if ________ is a tautology.
a) p ↔ q
b) p → q
c) ¬ (p ∨ q)
d) ¬p ∨ ¬q

Explanation: Definition of logical equivalence.

2. p → q is logically equivalent to ________
a) ¬p ∨ ¬q
b) p ∨ ¬q
c) ¬p ∨ q
d) ¬p ∧ q

Explanation: (p → q) ↔ (¬p ∨ q) is tautology.

3. p ∨ q is logically equivalent to ________
a) ¬q → ¬p
b) q → p
c) ¬p → ¬q
d) ¬p → q

Explanation: (p ∨ q) ↔ (¬p → q) is tautology.

4. ¬ (p ↔ q) is logically equivalent to ________
a) q↔p
b) p↔¬q
c) ¬p↔¬q
d) ¬q↔¬p

Explanation: ¬(p↔q)↔(p↔¬q) is tautology.

5. p ∧ q is logically equivalent to ________
a) ¬ (p → ¬q)
b) (p → ¬q)
c) (¬p → ¬q)
d) (¬p → q)

Explanation: (p ∧ q) ↔ (¬(p → ¬q)) is tautology.
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6. 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

Explanation: Verify using truth table, all are correct.

7. 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)

Explanation: (p ↔ q) ↔ ((p → q) ∧ (q → p)) is tautology.

8. (p → q) ∧ (p → r) is logically equivalent to ________
a) p → (q ∧ r)
b) p → (q ∨ r)
c) p ∧ (q ∨ r)
d) p ∨ (q ∧ r)

Explanation: ((p → q) ∧ (p → r)) ↔ (p → (q ∧ r)) is tautology.

9. (p → r) ∨ (q → r) is logically equivalent to ________
a) (p ∧ q) ∨ r
b) (p ∨ q) → r
c) (p ∧ q) → r
d) (p → q) → r

Explanation: ((p → r) ∨ (q → r)) ↔ ((p ∧ q) → r) is tautology.

10. ¬ (p ↔ q) is logically equivalent to ________
a) p ↔ ¬q
b) ¬p ↔ q
c) ¬p ↔ ¬q
d) ¬q ↔ ¬p

Explanation: (¬ (p ↔ q)) ↔ (p ↔ ¬q) is tautology.

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