Q.1 Which of the following is a tautology?
p ∨ ¬p
p ∧ ¬p
p → q
¬p → q
Explanation - The law of the excluded middle states that for any proposition p, either p or not p is always true, hence it is a tautology.
Correct answer is: p ∨ ¬p
Q.2 What is the contrapositive of the statement 'If it rains, then the ground is wet'?
If it does not rain, then the ground is wet
If the ground is wet, then it rains
If the ground is not wet, then it does not rain
If it rains, then the ground is not wet
Explanation - The contrapositive of 'if p then q' is 'if not q then not p'.
Correct answer is: If the ground is not wet, then it does not rain
Q.3 Which method of proof involves assuming the negation of the statement to be proven?
Direct proof
Proof by contraposition
Proof by contradiction
Mathematical induction
Explanation - Proof by contradiction assumes the negation of the statement and derives a contradiction, proving the original statement true.
Correct answer is: Proof by contradiction
Q.4 What is the truth value of (p → q) when p is true and q is false?
True
False
Cannot be determined
Sometimes true, sometimes false
Explanation - An implication p → q is false only when p is true and q is false.
Correct answer is: False
Q.5 Which of the following is logically equivalent to p → q?
¬p ∨ q
p ∨ q
¬p ∧ q
q → p
Explanation - The implication p → q is logically equivalent to ¬p ∨ q.
Correct answer is: ¬p ∨ q
Q.6 Which of the following is not a valid logical connective?
AND
OR
NOR
SUBTRACT
Explanation - Logical connectives include AND, OR, NOT, NAND, NOR, etc. SUBTRACT is not a logical connective.
Correct answer is: SUBTRACT
Q.7 In a truth table for two propositions, how many rows are needed?
2
3
4
8
Explanation - Each proposition has 2 truth values, so with 2 propositions we have 2² = 4 rows.
Correct answer is: 4
Q.8 Which rule of inference allows us to conclude q from p → q and p?
Modus Ponens
Modus Tollens
Disjunctive Syllogism
Hypothetical Syllogism
Explanation - Modus Ponens states: If p → q and p are true, then q must be true.
Correct answer is: Modus Ponens
Q.9 Which proof technique is commonly used for statements involving natural numbers?
Contradiction
Direct proof
Mathematical induction
Contrapositive
Explanation - Mathematical induction is widely used to prove statements involving natural numbers.
Correct answer is: Mathematical induction
Q.10 What is the converse of the implication p → q?
q → p
¬p → ¬q
¬q → ¬p
p ∨ q
Explanation - The converse of 'if p then q' is 'if q then p'.
Correct answer is: q → p
Q.11 What is the negation of the statement 'All dogs are friendly'?
Some dogs are not friendly
No dogs are friendly
All dogs are not friendly
Some dogs are friendly
Explanation - Negation of a universal statement 'All A are B' is 'Some A are not B'.
Correct answer is: Some dogs are not friendly
Q.12 What is the logical form of 'Either it rains or it snows'?
p ∧ q
p ∨ q
p → q
¬p ∧ q
Explanation - The word 'either...or' corresponds to logical disjunction (∨).
Correct answer is: p ∨ q
Q.13 Which statement is a contradiction?
p ∨ ¬p
p ∧ ¬p
p → p
p ↔ p
Explanation - p ∧ ¬p is always false regardless of the truth value of p, hence it is a contradiction.
Correct answer is: p ∧ ¬p
Q.14 Which of the following quantifiers means 'there exists at least one'?
∀
∃
⇒
↔
Explanation - The existential quantifier (∃) denotes 'there exists'.
Correct answer is: ∃
Q.15 What is the base case in mathematical induction?
The smallest value checked to start the proof
The largest value proven
The induction hypothesis
The induction step
Explanation - In induction, the base case verifies the statement for the initial value (usually n=1).
Correct answer is: The smallest value checked to start the proof
Q.16 If a statement is both a tautology and a contradiction, it must be:
Always true
Always false
Invalid
Impossible
Explanation - A statement cannot be both always true and always false, hence it is impossible.
Correct answer is: Impossible
Q.17 What is the negation of 'There exists an x such that P(x)'?
There exists no x such that P(x)
For all x, not P(x)
Some x does not satisfy P(x)
For some x, P(x) is false
Explanation - Negation of ∃x P(x) is ∀x ¬P(x).
Correct answer is: For all x, not P(x)
Q.18 In a truth table for 3 propositions, how many rows are needed?
4
6
8
12
Explanation - With 3 propositions, there are 2³ = 8 rows in the truth table.
Correct answer is: 8
Q.19 Which proof technique assumes a property holds for n=k and then proves for n=k+1?
Direct proof
Contradiction
Contrapositive
Mathematical induction
Explanation - The induction step assumes n=k (induction hypothesis) and proves n=k+1.
Correct answer is: Mathematical induction
Q.20 Which of the following is a valid use of Modus Tollens?
p → q, p ⟹ q
p → q, ¬q ⟹ ¬p
p ∨ q, ¬p ⟹ q
p → q, q ⟹ p
Explanation - Modus Tollens: From p → q and ¬q, we conclude ¬p.
Correct answer is: p → q, ¬q ⟹ ¬p
Q.21 Which of the following is an example of a universal quantifier?
All humans are mortal
Some humans are tall
There exists a prime number greater than 100
At least one bird can fly
Explanation - Universal quantifier (∀) expresses that a property holds for all elements of a set.
Correct answer is: All humans are mortal
Q.22 Which logical law is represented by (p ∧ q) → p?
Law of identity
Law of simplification
Law of excluded middle
Law of contradiction
Explanation - The simplification law states that from p ∧ q, we can infer p.
Correct answer is: Law of simplification
Q.23 What is an axiom?
A proven theorem
A self-evident truth
A contradictory statement
A logical fallacy
Explanation - An axiom is a statement taken to be true without proof.
Correct answer is: A self-evident truth
Q.24 What is the result of double negation law?
¬(¬p) is equivalent to p
¬(¬p) is equivalent to ¬p
¬(¬p) is equivalent to false
¬(¬p) is equivalent to true
Explanation - The law of double negation states that negating a negation gives back the original proposition.
Correct answer is: ¬(¬p) is equivalent to p
Q.25 If two statements are logically equivalent, their truth tables are:
Identical
Opposite
Contradictory
Irrelevant
Explanation - Two propositions are logically equivalent if they have identical truth values for all cases.
Correct answer is: Identical