WGU C960 Discrete Mathematics
Exam 2025/2026 – Verified Q&A |
100% Correct Answers
Logic (Questions 1–40)
1. Which of the following is a tautology? A. p ∧ q B. ¬(p ∧ q) ↔ (¬p ∨ ¬q) C. p → q D. p ↔ q
Explanation: This is De Morgan's law, which is always true regardless of truth values, making it
a tautology. In logic, it demonstrates equivalence in negation distribution, useful for simplifying
expressions and proving validity in propositional logic.
2. The converse of "If it rains, then the ground is wet" is: A. If the ground is wet, then it rains
B. If the ground is wet, then it rains C. If it does not rain, then the ground is not wet D. If the
ground is not wet, then it does not rain
Explanation: The converse reverses antecedent and consequent (q → p). In logic, the converse
is not necessarily true, distinguishing it from the original implication for conditional reasoning
and biconditional statements.
3. A contradiction is a proposition that: A. Is true for some assignments B. Is false for all truth
assignments C. Is equivalent to p ∨ ¬p D. Has undefined truth value
Explanation: Contradictions like p ∧ ¬p are always false, essential for identifying invalid
arguments in propositional logic via truth tables.
4. The proposition (p → q) ∧ (q → p) is logically equivalent to: A. p ∧ q B. p ↔ q C. p ∨ q D.
¬p ∧ ¬q
Explanation: This biconditional equivalence holds when both directions are true. In logic, it
models mutual implications, used in equivalence proofs.
5. In a truth table for (p ∨ q) → r, how many rows are there? A. 2 B. 8 C. 4 D. 16
Explanation: Three variables yield 2^3 = 8 rows. Truth tables systematically evaluate
propositional logic expressions for tautologies or contradictions.
6. Which is logically equivalent to ¬(p ∨ q)? A. ¬p ∨ ¬q B. ¬p ∧ ¬q C. p ∧ q D. p ∨ ¬q
, Explanation: De Morgan's law: negation distributes over disjunction as conjunction of
negations, a fundamental identity in Boolean algebra and logic simplification.
7. The inverse of p → q is: A. ¬p → ¬q B. ¬p → q C. q → ¬p D. ¬q → p
Explanation: Inverse negates antecedent (¬p → q). In logic, neither inverse nor converse
preserves truth value of the original conditional.
8. A valid argument has: A. True conclusion B. True conclusion when premises are true C.
False premises D. Equivalent premises
Explanation: Validity in propositional logic means the conclusion follows from premises; truth
tables or rules like modus ponens verify.
9. Modus tollens is: A. p → q, p, therefore q B. p → q, ¬q, therefore ¬p C. p ∧ q, therefore p D.
p ∨ q, ¬p, therefore q
Explanation: This rule denies the consequent to deny antecedent. It's a core inference in
deductive logic for argument soundness.
10. The proposition p ≡ (p ∧ q) ∨ (p ∧ ¬q) simplifies to: A. p ∨ q B. p C. q D. p ∧ q
Explanation: Distributive law shows p factors out, a tautology illustrating idempotence in
propositional logic.
11. Which is a contingent proposition? A. p ∧ ¬p B. p ∧ q C. p ∨ ¬p D. ¬(p ∨ q) ∧ (p ∧ q)
Explanation: Contingent statements are true/false depending on assignments, unlike
tautologies/contradictions; useful for analyzing argument validity.
12. The contrapositive of p → q is: A. q → p B. ¬q → ¬p C. ¬p → q D. ¬p ∨ q
Explanation: Contrapositive is logically equivalent, preserving truth; key for conditional proofs
in logic.
13. In predicate logic, ∀x (P(x) → Q(x)) means: A. Some x satisfy P and Q B. Every x with P
has Q C. All x have P or Q D. No x has P and Q
Explanation: Universal quantifier with implication; used in formal proofs for "all elements
satisfying condition."
14. ∃x ∀y (R(x,y)) translates to: A. For all y, there exists x such that R B. There exists x such
that for all y, R(x,y) C. For all x, some y R D. Some x, some y R
Explanation: Quantifier order matters; this asserts one x relates to every y, critical in relational
logic.
Exam 2025/2026 – Verified Q&A |
100% Correct Answers
Logic (Questions 1–40)
1. Which of the following is a tautology? A. p ∧ q B. ¬(p ∧ q) ↔ (¬p ∨ ¬q) C. p → q D. p ↔ q
Explanation: This is De Morgan's law, which is always true regardless of truth values, making it
a tautology. In logic, it demonstrates equivalence in negation distribution, useful for simplifying
expressions and proving validity in propositional logic.
2. The converse of "If it rains, then the ground is wet" is: A. If the ground is wet, then it rains
B. If the ground is wet, then it rains C. If it does not rain, then the ground is not wet D. If the
ground is not wet, then it does not rain
Explanation: The converse reverses antecedent and consequent (q → p). In logic, the converse
is not necessarily true, distinguishing it from the original implication for conditional reasoning
and biconditional statements.
3. A contradiction is a proposition that: A. Is true for some assignments B. Is false for all truth
assignments C. Is equivalent to p ∨ ¬p D. Has undefined truth value
Explanation: Contradictions like p ∧ ¬p are always false, essential for identifying invalid
arguments in propositional logic via truth tables.
4. The proposition (p → q) ∧ (q → p) is logically equivalent to: A. p ∧ q B. p ↔ q C. p ∨ q D.
¬p ∧ ¬q
Explanation: This biconditional equivalence holds when both directions are true. In logic, it
models mutual implications, used in equivalence proofs.
5. In a truth table for (p ∨ q) → r, how many rows are there? A. 2 B. 8 C. 4 D. 16
Explanation: Three variables yield 2^3 = 8 rows. Truth tables systematically evaluate
propositional logic expressions for tautologies or contradictions.
6. Which is logically equivalent to ¬(p ∨ q)? A. ¬p ∨ ¬q B. ¬p ∧ ¬q C. p ∧ q D. p ∨ ¬q
, Explanation: De Morgan's law: negation distributes over disjunction as conjunction of
negations, a fundamental identity in Boolean algebra and logic simplification.
7. The inverse of p → q is: A. ¬p → ¬q B. ¬p → q C. q → ¬p D. ¬q → p
Explanation: Inverse negates antecedent (¬p → q). In logic, neither inverse nor converse
preserves truth value of the original conditional.
8. A valid argument has: A. True conclusion B. True conclusion when premises are true C.
False premises D. Equivalent premises
Explanation: Validity in propositional logic means the conclusion follows from premises; truth
tables or rules like modus ponens verify.
9. Modus tollens is: A. p → q, p, therefore q B. p → q, ¬q, therefore ¬p C. p ∧ q, therefore p D.
p ∨ q, ¬p, therefore q
Explanation: This rule denies the consequent to deny antecedent. It's a core inference in
deductive logic for argument soundness.
10. The proposition p ≡ (p ∧ q) ∨ (p ∧ ¬q) simplifies to: A. p ∨ q B. p C. q D. p ∧ q
Explanation: Distributive law shows p factors out, a tautology illustrating idempotence in
propositional logic.
11. Which is a contingent proposition? A. p ∧ ¬p B. p ∧ q C. p ∨ ¬p D. ¬(p ∨ q) ∧ (p ∧ q)
Explanation: Contingent statements are true/false depending on assignments, unlike
tautologies/contradictions; useful for analyzing argument validity.
12. The contrapositive of p → q is: A. q → p B. ¬q → ¬p C. ¬p → q D. ¬p ∨ q
Explanation: Contrapositive is logically equivalent, preserving truth; key for conditional proofs
in logic.
13. In predicate logic, ∀x (P(x) → Q(x)) means: A. Some x satisfy P and Q B. Every x with P
has Q C. All x have P or Q D. No x has P and Q
Explanation: Universal quantifier with implication; used in formal proofs for "all elements
satisfying condition."
14. ∃x ∀y (R(x,y)) translates to: A. For all y, there exists x such that R B. There exists x such
that for all y, R(x,y) C. For all x, some y R D. Some x, some y R
Explanation: Quantifier order matters; this asserts one x relates to every y, critical in relational
logic.
