20 Mathematical Logic Quiz Questions and Answers

1. Question: What is the truth value of the proposition \( p \land q \) if \( p \) is true and \( q \) is false?
Options:
A) True
B) False
C) Cannot be determined
D) True only if both are true
Answer: B) False
Explanation: The conjunction \( p \land q \) is true only when both \( p \) and \( q \) are true. Since \( p \) is true and \( q \) is false, the statement is false.

2. Question: Which of the following is logically equivalent to \( \neg (p \lor q) \)?
Options:
A) \( \neg p \lor q \)
B) \( \neg p \land \neg q \)
C) \( p \land q \)
D) \( \neg p \lor \neg q \)
Answer: B) \( \neg p \land \neg q \)
Explanation: This is De Morgan’s law, which states that the negation of a disjunction is the conjunction of the negations.

3. Question: If \( p \rightarrow q \) is true and \( q \) is false, what must be the truth value of \( p \)?
Options:
A) True
B) False
C) Cannot be determined
D) True only if q is true
Answer: B) False
Explanation: In an implication \( p \rightarrow q \), if \( q \) is false, then \( p \) must be false for the implication to be true.

4. Question: What is the contrapositive of the statement “If it is raining, then the streets are wet”?
Options:
A) If the streets are wet, then it is raining
B) If the streets are not wet, then it is not raining
C) If it is not raining, then the streets are not wet
D) It is raining and the streets are wet
Answer: B) If the streets are not wet, then it is not raining
Explanation: The contrapositive of \( p \rightarrow q \) is \( \neg q \rightarrow \neg p \), which preserves the logical equivalence.

5. Question: Which of the following statements is a tautology?
Options:
A) \( p \land \neg p \)
B) \( p \lor \neg p \)
C) \( p \rightarrow p \)
D) Both B and C
Answer: D) Both B and C
Explanation: \( p \lor \neg p \) is the law of excluded middle, and \( p \rightarrow p \) is always true, making both tautologies.

6. Question: In predicate logic, what does \( \forall x (P(x) \rightarrow Q(x)) \) mean?
Options:
A) For all x, P(x) and Q(x) are true
B) There exists x such that P(x) implies Q(x)
C) For all x, if P(x) is true, then Q(x) is true
D) For some x, P(x) and Q(x) are false
Answer: C) For all x, if P(x) is true, then Q(x) is true
Explanation: The universal quantifier \( \forall \) means the implication holds for every x.

7. Question: If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∩ B?
Options:
A) {1, 2, 3, 4, 5}
B) {1, 2, 3}
C) {3}
D) {1, 2, 4, 5}
Answer: C) {3}
Explanation: The intersection of two sets contains only the elements common to both, which is {3}.

8. Question: Which rule of inference is used in: “If p, then q. p is true. Therefore, q is true”?
Options:
A) Modus tollens
B) Modus ponens
C) Disjunctive syllogism
D) Hypothetical syllogism
Answer: B) Modus ponens
Explanation: Modus ponens states that from \( p \rightarrow q \) and p, we can conclude q.

9. Question: What is the truth value of \( (p \rightarrow q) \land (q \rightarrow p) \) when p is true and q is false?
Options:
A) True
B) False
C) Cannot be determined
D) True if p and q are the same
Answer: B) False
Explanation: This is a biconditional equivalence. Since p is true and q is false, both implications are false, making the conjunction false.

10. Question: In set theory, if A ⊆ B and B ⊆ A, what is the relationship between A and B?
Options:
A) A is a proper subset of B
B) A and B are equal
C) A and B are disjoint
D) B is a subset of A
Answer: B) A and B are equal
Explanation: If every element of A is in B and every element of B is in A, then A equals B.

11. Question: What does \( \exists x P(x) \) mean?
Options:
A) For all x, P(x) is true
B) There exists at least one x such that P(x) is true
C) For no x is P(x) true
D) P(x) is true for exactly one x
Answer: B) There exists at least one x such that P(x) is true
Explanation: The existential quantifier \( \exists \) indicates that there is at least one instance where the predicate holds.

12. Question: If p is false, what is the truth value of \( p \lor q \)?
Options:
A) True
B) False
C) Depends on q
D) Always true
Answer: C) Depends on q
Explanation: The disjunction \( p \lor q \) is true if q is true and false if q is false, since p is false.

13. Question: Which of the following is a contradiction?
Options:
A) \( p \lor \neg p \)
B) \( p \land \neg p \)
C) \( p \rightarrow \neg p \) if p is true
D) Both B and C
Answer: D) Both B and C
Explanation: A contradiction is always false; \( p \land \neg p \) is false for any p, and \( p \rightarrow \neg p \) is false when p is true.

14. Question: What is the fallacy in: “If it rains, the game is canceled. The game is canceled. Therefore, it rains”?
Options:
A) Affirming the consequent
B) Denying the antecedent
C) Modus ponens
D) Valid argument
Answer: A) Affirming the consequent
Explanation: This incorrectly assumes that because q is true, p must be true in an implication p → q.

15. Question: For sets A = {1, 2} and B = {2, 3}, what is A ∪ B?
Options:
A) {1, 2}
B) {2, 3}
C) {1, 2, 3}
D) {1, 3}
Answer: C) {1, 2, 3}
Explanation: The union of two sets contains all unique elements from both sets.

16. Question: In logic, \( p \leftrightarrow q \) is equivalent to:
Options:
A) \( (p \rightarrow q) \land (q \rightarrow p) \)
B) \( p \land q \)
C) \( p \lor q \)
D) \( \neg p \lor \neg q \)
Answer: A) \( (p \rightarrow q) \land (q \rightarrow p) \)
Explanation: The biconditional means p implies q and q implies p.

17. Question: If no students are present, what is the truth value of \( \forall x \text{ Student}(x) \rightarrow \text{Present}(x) \)?
Options:
A) True
B) False
C) Cannot be determined
D) False if students exist
Answer: A) True
Explanation: The implication is true because the antecedent is false for all x (no students), making the universal statement true.

18. Question: What is the negation of \( \forall x P(x) \)?
Options:
A) \( \forall x \neg P(x) \)
B) \( \exists x \neg P(x) \)
C) \( \neg \forall x P(x) \)
D) Both A and B
Answer: B) \( \exists x \neg P(x) \)
Explanation: The negation of a universal quantifier is an existential quantifier with the negated predicate.

19. Question: In a truth table, how many rows are needed for three propositions p, q, and r?
Options:
A) 4
B) 8
C) 6
D) 3
Answer: B) 8
Explanation: For n propositions, there are 2^n rows; for three propositions, it is 2^3 = 8 rows.

20. Question: Which of the following is an example of denying the antecedent?
Options:
A) If p, then q. q is true. Therefore, p is true.
B) If p, then q. Not p. Therefore, not q.
C) If p, then q. Not q. Therefore, not p.
D) If p, then q. p is true. Therefore, q is true.
Answer: B) If p, then q. Not p. Therefore, not q.
Explanation: Denying the antecedent fallaciously concludes that if p is false, then q must be false in an implication p → q.