WGU MAT 215: DISCRETE MATHEMATICS COMPLETE EXAM |QUESTIONS
AND WELL-EXPLAINED VERIFIED ANSWERS GRADED A+| 2026/2027
Q1. Which of the following is a proposition?
A) The sky is blue. ✓
B) What time is it?
C) Open the door.
D) Please sit down.
Explanation: A proposition is a declarative statement that is either true or
false. Only 'The sky is blue' qualifies.
Q2. What is the negation of 'It is raining'?
A) It is not raining. ✓
B) It might be raining.
C) It will rain.
D) Rain is likely.
Explanation: Negation (¬p) of 'It is raining' is 'It is not raining.'
Q3. The compound proposition p ∧ q is true when:
A) p is true and q is false.
B) Both p and q are true. ✓
C) p is false and q is true.
D) Both p and q are false.
Explanation: Conjunction (AND) is true only when both operands are true.
Q4. Which logical connective represents 'or'?
A) ∧
B) ¬
C) ∨ ✓
D) →
Explanation: The symbol ∨ represents disjunction (inclusive or).
Q5. A tautology is a compound proposition that is:
WGU MAT 215 Discrete Mathematics — Study Guide
, A) Always false.
B) Sometimes true.
C) Always true. ✓
D) Undefined.
Explanation: A tautology has truth value TRUE for all combinations of its
variables.
Q6. Which of the following is logically equivalent to ¬(p ∧ q)?
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q ✓
C) p ∨ q
D) p ∧ q
Explanation: By De Morgan's Law: ¬(p ∧ q) ≡ ¬p ∨ ¬q.
Q7. The conditional p → q is FALSE only when:
A) p is false and q is true.
B) p is true and q is true.
C) p is false and q is false.
D) p is true and q is false. ✓
Explanation: An implication is false only when the hypothesis is true but
the conclusion is false.
Q8. What is the contrapositive of p → q?
A) q → p
B) ¬p → ¬q
C) ¬q → ¬p ✓
D) ¬p → q
Explanation: The contrapositive of p → q is ¬q → ¬p, which is logically
equivalent to the original.
Q9. p ↔ q (biconditional) is TRUE when:
A) p and q have opposite truth values.
B) p and q have the same truth value. ✓
C) p is true.
D) q is false.
Explanation: A biconditional is true exactly when both sides share the
same truth value.
Q10. Which of the following is a contradiction?
A) p ∨ ¬p
B) p ∧ ¬p ✓
WGU MAT 215 Discrete Mathematics — Study Guide
, C) p → p
D) p ↔ p
Explanation: p ∧ ¬p is always false — a contradiction — since p cannot be
both true and false.
Q11. The converse of p → q is:
A) ¬p → ¬q
B) q → p ✓
C) ¬q → ¬p
D) p → ¬q
Explanation: The converse swaps the hypothesis and conclusion: q → p.
Q12. Logical equivalence p ≡ q means:
A) p and q have the same truth table. ✓
B) p implies q.
C) p is a tautology.
D) q is a contradiction.
Explanation: Two propositions are logically equivalent when they produce
identical truth tables.
Q13. Which law states p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)?
A) Associative Law
B) Commutative Law
C) Distributive Law ✓
D) Absorption Law
Explanation: The Distributive Law distributes ∨ over ∧.
Q14. What does the symbol ∀ represent?
A) There exists
B) For all (universal quantifier) ✓
C) Such that
D) Negation
Explanation: ∀ is the universal quantifier meaning 'for all' or 'for every.'
Q15. What does the symbol ∃ represent?
A) For all
B) Negation
C) There exists (existential quantifier) ✓
D) Implication
Explanation: ∃ is the existential quantifier meaning 'there exists at least
one.'
WGU MAT 215 Discrete Mathematics — Study Guide
, Q16. The negation of ∀x P(x) is:
A) ∀x ¬P(x)
B) ∃x P(x)
C) ∃x ¬P(x) ✓
D) ¬∀x ¬P(x)
Explanation: ¬(∀x P(x)) ≡ ∃x ¬P(x) — there exists at least one x for which
P(x) is false.
Q17. Which argument form is Modus Ponens?
A) p → q, ¬p ∴ ¬q
B) p → q, p ∴ q ✓
C) p → q, q ∴ p
D) ¬q → ¬p, p ∴ q
Explanation: Modus Ponens: if p → q and p are both true, then q must be
true.
Q18. Modus Tollens has the form:
A) p → q, q ∴ p
B) p → q, p ∴ q
C) p → q, ¬q ∴ ¬p ✓
D) ¬p → ¬q, ¬q ∴ ¬p
Explanation: Modus Tollens: if p → q and ¬q, then ¬p follows.
Q19. p ∨ T ≡ ?
A) p
B) F
C) T ✓
D) ¬p
Explanation: By the Domination Law, p ∨ T is always TRUE.
Q20. p ∧ F ≡ ?
A) p
B) T
C) F ✓
D) ¬p
Explanation: By the Domination Law, p ∧ F is always FALSE.
Q21. Which of the following represents exclusive or (XOR)?
A) p ↔ q
B) (p ∨ q) ∧ ¬(p ∧ q) ✓
C) p ∧ q
WGU MAT 215 Discrete Mathematics — Study Guide
AND WELL-EXPLAINED VERIFIED ANSWERS GRADED A+| 2026/2027
Q1. Which of the following is a proposition?
A) The sky is blue. ✓
B) What time is it?
C) Open the door.
D) Please sit down.
Explanation: A proposition is a declarative statement that is either true or
false. Only 'The sky is blue' qualifies.
Q2. What is the negation of 'It is raining'?
A) It is not raining. ✓
B) It might be raining.
C) It will rain.
D) Rain is likely.
Explanation: Negation (¬p) of 'It is raining' is 'It is not raining.'
Q3. The compound proposition p ∧ q is true when:
A) p is true and q is false.
B) Both p and q are true. ✓
C) p is false and q is true.
D) Both p and q are false.
Explanation: Conjunction (AND) is true only when both operands are true.
Q4. Which logical connective represents 'or'?
A) ∧
B) ¬
C) ∨ ✓
D) →
Explanation: The symbol ∨ represents disjunction (inclusive or).
Q5. A tautology is a compound proposition that is:
WGU MAT 215 Discrete Mathematics — Study Guide
, A) Always false.
B) Sometimes true.
C) Always true. ✓
D) Undefined.
Explanation: A tautology has truth value TRUE for all combinations of its
variables.
Q6. Which of the following is logically equivalent to ¬(p ∧ q)?
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q ✓
C) p ∨ q
D) p ∧ q
Explanation: By De Morgan's Law: ¬(p ∧ q) ≡ ¬p ∨ ¬q.
Q7. The conditional p → q is FALSE only when:
A) p is false and q is true.
B) p is true and q is true.
C) p is false and q is false.
D) p is true and q is false. ✓
Explanation: An implication is false only when the hypothesis is true but
the conclusion is false.
Q8. What is the contrapositive of p → q?
A) q → p
B) ¬p → ¬q
C) ¬q → ¬p ✓
D) ¬p → q
Explanation: The contrapositive of p → q is ¬q → ¬p, which is logically
equivalent to the original.
Q9. p ↔ q (biconditional) is TRUE when:
A) p and q have opposite truth values.
B) p and q have the same truth value. ✓
C) p is true.
D) q is false.
Explanation: A biconditional is true exactly when both sides share the
same truth value.
Q10. Which of the following is a contradiction?
A) p ∨ ¬p
B) p ∧ ¬p ✓
WGU MAT 215 Discrete Mathematics — Study Guide
, C) p → p
D) p ↔ p
Explanation: p ∧ ¬p is always false — a contradiction — since p cannot be
both true and false.
Q11. The converse of p → q is:
A) ¬p → ¬q
B) q → p ✓
C) ¬q → ¬p
D) p → ¬q
Explanation: The converse swaps the hypothesis and conclusion: q → p.
Q12. Logical equivalence p ≡ q means:
A) p and q have the same truth table. ✓
B) p implies q.
C) p is a tautology.
D) q is a contradiction.
Explanation: Two propositions are logically equivalent when they produce
identical truth tables.
Q13. Which law states p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)?
A) Associative Law
B) Commutative Law
C) Distributive Law ✓
D) Absorption Law
Explanation: The Distributive Law distributes ∨ over ∧.
Q14. What does the symbol ∀ represent?
A) There exists
B) For all (universal quantifier) ✓
C) Such that
D) Negation
Explanation: ∀ is the universal quantifier meaning 'for all' or 'for every.'
Q15. What does the symbol ∃ represent?
A) For all
B) Negation
C) There exists (existential quantifier) ✓
D) Implication
Explanation: ∃ is the existential quantifier meaning 'there exists at least
one.'
WGU MAT 215 Discrete Mathematics — Study Guide
, Q16. The negation of ∀x P(x) is:
A) ∀x ¬P(x)
B) ∃x P(x)
C) ∃x ¬P(x) ✓
D) ¬∀x ¬P(x)
Explanation: ¬(∀x P(x)) ≡ ∃x ¬P(x) — there exists at least one x for which
P(x) is false.
Q17. Which argument form is Modus Ponens?
A) p → q, ¬p ∴ ¬q
B) p → q, p ∴ q ✓
C) p → q, q ∴ p
D) ¬q → ¬p, p ∴ q
Explanation: Modus Ponens: if p → q and p are both true, then q must be
true.
Q18. Modus Tollens has the form:
A) p → q, q ∴ p
B) p → q, p ∴ q
C) p → q, ¬q ∴ ¬p ✓
D) ¬p → ¬q, ¬q ∴ ¬p
Explanation: Modus Tollens: if p → q and ¬q, then ¬p follows.
Q19. p ∨ T ≡ ?
A) p
B) F
C) T ✓
D) ¬p
Explanation: By the Domination Law, p ∨ T is always TRUE.
Q20. p ∧ F ≡ ?
A) p
B) T
C) F ✓
D) ¬p
Explanation: By the Domination Law, p ∧ F is always FALSE.
Q21. Which of the following represents exclusive or (XOR)?
A) p ↔ q
B) (p ∨ q) ∧ ¬(p ∧ q) ✓
C) p ∧ q
WGU MAT 215 Discrete Mathematics — Study Guide
