Q.1 Which of the following is a tautology?
p ∧ ¬p
p ∨ ¬p
p → ¬p
¬p ∨ p ∧ q
Explanation - A tautology is always true regardless of truth values. 'p ∨ ¬p' is always true (Law of Excluded Middle).
Correct answer is: p ∨ ¬p
Q.2 What is the contrapositive of 'If it rains, then the ground is wet'?
If the ground is not wet, then it did not rain
If it rains, then the ground is not wet
If the ground is wet, then it rained
It did not rain if the ground is not wet
Explanation - Contrapositive of 'p → q' is '¬q → ¬p'.
Correct answer is: If the ground is not wet, then it did not rain
Q.3 Which of the following represents De Morgan’s law?
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p ∨ q) ≡ ¬p ∨ ¬q
¬(p ∧ q) ≡ ¬p ∧ ¬q
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
Explanation - De Morgan’s laws state: ¬(p ∨ q) ≡ ¬p ∧ ¬q and ¬(p ∧ q) ≡ ¬p ∨ ¬q.
Correct answer is: ¬(p ∨ q) ≡ ¬p ∧ ¬q
Q.4 The statement 'All prime numbers are odd' is:
True
False
Sometimes true
Cannot be determined
Explanation - The statement is false because 2 is prime but even.
Correct answer is: False
Q.5 Which of the following is logically equivalent to 'p → q'?
¬p ∨ q
p ∧ q
q → p
p ∨ ¬q
Explanation - 'p → q' is equivalent to '¬p ∨ q'.
Correct answer is: ¬p ∨ q
Q.6 The negation of 'Some cats are black' is:
No cats are black
All cats are black
Some cats are not black
All cats are not black
Explanation - Negation of 'Some A are B' is 'No A are B'.
Correct answer is: No cats are black
Q.7 Which proof technique assumes the negation of the statement to be proved?
Direct proof
Proof by induction
Proof by contradiction
Proof by construction
Explanation - Proof by contradiction assumes ¬P and shows it leads to a contradiction.
Correct answer is: Proof by contradiction
Q.8 If 'p' is false and 'q' is true, what is the truth value of 'p → q'?
True
False
Cannot be determined
Sometimes true
Explanation - Implication 'p → q' is true when p is false, regardless of q.
Correct answer is: True
Q.9 The converse of 'If it is raining, then the ground is wet' is:
If it is not raining, then the ground is not wet
If the ground is wet, then it is raining
If it is raining, then the ground is not wet
If the ground is not wet, then it is not raining
Explanation - Converse of 'p → q' is 'q → p'.
Correct answer is: If the ground is wet, then it is raining
Q.10 Which of the following is an example of a proposition?
What time is it?
x + 2 = 5
The sky is blue
Come here!
Explanation - A proposition is a declarative sentence with a truth value. 'The sky is blue' is either true or false.
Correct answer is: The sky is blue
Q.11 Which logical connective represents 'and'?
∨
→
∧
¬
Explanation - '∧' denotes conjunction (logical AND).
Correct answer is: ∧
Q.12 What is the negation of 'If it rains, then I will stay home'?
If it rains, I will not stay home
It rains and I will not stay home
It does not rain and I will stay home
I will stay home only if it rains
Explanation - Negation of 'p → q' is 'p ∧ ¬q'.
Correct answer is: It rains and I will not stay home
Q.13 Which of the following is NOT a valid proof method?
Proof by induction
Proof by counterexample
Proof by assumption
Proof by contradiction
Explanation - Valid proof methods are direct, induction, contradiction, contrapositive, and counterexample. 'Proof by assumption' is not standard.
Correct answer is: Proof by assumption
Q.14 What is the truth value of '¬(T ∨ F)'?
True
False
Both
Undefined
Explanation - T ∨ F = T, so ¬(T) = F.
Correct answer is: False
Q.15 Which of the following is a contradiction?
p ∧ ¬p
p ∨ q
p ∧ q
p → q
Explanation - A contradiction is always false. 'p ∧ ¬p' is always false.
Correct answer is: p ∧ ¬p
Q.16 In logic, 'iff' stands for:
If and for fun
If and only if
If false
Infinite function form
Explanation - 'iff' means biconditional: p ↔ q.
Correct answer is: If and only if
Q.17 Which of the following is a valid use of proof by induction?
To prove a formula for sum of first n integers
To prove 2 is prime
To prove the Earth is round
To prove existence of irrational numbers
Explanation - Induction is used for statements involving natural numbers.
Correct answer is: To prove a formula for sum of first n integers
Q.18 The biconditional 'p ↔ q' is equivalent to:
(p ∧ q) ∨ (¬p ∧ ¬q)
(p ∧ ¬q) ∨ (¬p ∧ q)
p ∨ q
¬p ∨ q
Explanation - Biconditional is true when both have same truth values.
Correct answer is: (p ∧ q) ∨ (¬p ∧ ¬q)
Q.19 Which law allows us to simplify 'p ∨ (p ∧ q)' into 'p'?
Idempotent law
Absorption law
Associative law
Distributive law
Explanation - Absorption law: p ∨ (p ∧ q) ≡ p.
Correct answer is: Absorption law
Q.20 What is the truth value of 'p ∧ q' if p = T and q = F?
True
False
Both
Undefined
Explanation - Conjunction is true only when both are true.
Correct answer is: False
Q.21 Which quantifier denotes 'there exists'?
∀
∃
¬
→
Explanation - ∃ denotes existence.
Correct answer is: ∃
Q.22 The statement 'If n is even, then n² is even' is best proved by:
Contradiction
Direct proof
Counterexample
Induction
Explanation - Assume n = 2k, then n² = 4k² = 2(2k²), so even.
Correct answer is: Direct proof
Q.23 Which of the following is equivalent to '¬(p ∧ q)'?
¬p ∨ ¬q
¬p ∧ ¬q
p ∧ ¬q
¬p ∨ q
Explanation - By De Morgan’s law: ¬(p ∧ q) ≡ ¬p ∨ ¬q.
Correct answer is: ¬p ∨ ¬q
Q.24 Which proof technique is used to prove statements like 'All swans are white'?
Direct proof
Proof by contradiction
Proof by counterexample
Proof by induction
Explanation - To disprove a universal claim, provide a single counterexample.
Correct answer is: Proof by counterexample
Q.25 What is the negation of '∀x, P(x)'?
∃x, P(x)
∃x, ¬P(x)
¬∃x, P(x)
∀x, ¬P(x)
Explanation - Negating 'for all' changes to 'there exists' with negated predicate.
Correct answer is: ∃x, ¬P(x)