WGU MAT 215: DISCRETE MATHEMATICS COMPREHENSIVE
EXAM Q&A - VERIFIED ANSWERS - LATEST EDITION -
COMPLETE RESOURCE (2026/2027)
Q1 What is a proposition?
ANS A proposition is a declarative statement that is either true or false, but
not both.
Q2 What is a tautology?
ANS A tautology is a compound proposition that is always true, regardless of
the truth values of its components (e.g., p ∨ ¬p).
Q3 What is a contradiction?
ANS A contradiction is a compound proposition that is always false,
regardless of the truth values of its components (e.g., p ∧ ¬p).
Q4 What is a contingency?
ANS A contingency is a proposition that is neither a tautology nor a
contradiction — it is true for some assignments and false for others.
Q5 Define logical conjunction (AND).
ANS The conjunction of p and q, written p ∧ q, is true only when both p and
q are true.
Q6 Define logical disjunction (OR).
ANS The disjunction of p and q, written p ∨ q, is true when at least one of p
or q is true.
,Q7 Define logical negation (NOT).
ANS The negation of p, written ¬p, is true when p is false, and false when p
is true.
Q8 What is the conditional (implication) p → q?
ANS The conditional p → q is false only when p is true and q is false; it is
true in all other cases.
Q9 What is the biconditional p ↔ q?
ANS The biconditional p ↔ q is true when p and q have the same truth value
(both true or both false).
Q10 What is the contrapositive of p → q?
ANS The contrapositive is ¬q → ¬p. It is logically equivalent to the original
conditional.
Q11 What is the converse of p → q?
ANS The converse is q → p. It is NOT logically equivalent to the original
conditional.
Q12 What is the inverse of p → q?
ANS The inverse is ¬p → ¬q. It is logically equivalent to the converse but
NOT to the original conditional.
Q13 What does it mean for two propositions to be logically equivalent?
ANS Two propositions are logically equivalent if they have the same truth
value for every possible assignment of truth values to their variables,
written P ≡ Q.
Q14 State De Morgan's first law.
,ANS ¬(p ∧ q) ≡ ¬p ∨ ¬q. The negation of a conjunction equals the
disjunction of the negations.
Q15 State De Morgan's second law.
ANS ¬(p ∨ q) ≡ ¬p ∧ ¬q. The negation of a disjunction equals the
conjunction of the negations.
Q16 What is exclusive or (XOR)?
ANS The exclusive or of p and q, written p ⊕ q, is true when exactly one of
p or q is true, but not both.
Q17 What is modus ponens?
ANS Modus ponens: Given p → q and p are true, we conclude q is true. It is
a fundamental rule of inference.
Q18 What is modus tollens?
ANS Modus tollens: Given p → q and ¬q are true, we conclude ¬p is true.
Q19 What is the absorption law for OR?
ANS p ∨ (p ∧ q) ≡ p. Absorbing a conjunction into a disjunction leaves p
unchanged.
Q20 What is the distributive law of AND over OR?
ANS p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).
2. Predicates and Quantifiers
Q21 What is a predicate?
ANS A predicate is a function that takes one or more variables and returns a
proposition. Example: P(x) = 'x is even.'
, Q22 What is the universal quantifier?
ANS The universal quantifier ∀ means 'for all.' ∀x P(x) asserts that P(x) is
true for every element x in the domain.
Q23 What is the existential quantifier?
ANS The existential quantifier ∃ means 'there exists.' ∃x P(x) asserts that
P(x) is true for at least one element x in the domain.
Q24 What is the negation of ∀x P(x)?
ANS ¬(∀x P(x)) ≡ ∃x ¬P(x). There exists at least one x for which P(x) is
false.
Q25 What is the negation of ∃x P(x)?
ANS ¬(∃x P(x)) ≡ ∀x ¬P(x). P(x) is false for every x in the domain.
Q26 What is a free variable in a predicate?
ANS A free variable is a variable in a predicate that is not bound by any
quantifier.
Q27 What is a bound variable?
ANS A bound variable is a variable that has been quantified (bound by ∀ or
∃) within a logical expression.
Q28 What is the domain (universe of discourse)?
ANS The domain is the set of all possible values that variables in a predicate
can take.
Q29 What is a counterexample in logic?
ANS A counterexample is a specific value of the variable that makes a
universally quantified statement false.
EXAM Q&A - VERIFIED ANSWERS - LATEST EDITION -
COMPLETE RESOURCE (2026/2027)
Q1 What is a proposition?
ANS A proposition is a declarative statement that is either true or false, but
not both.
Q2 What is a tautology?
ANS A tautology is a compound proposition that is always true, regardless of
the truth values of its components (e.g., p ∨ ¬p).
Q3 What is a contradiction?
ANS A contradiction is a compound proposition that is always false,
regardless of the truth values of its components (e.g., p ∧ ¬p).
Q4 What is a contingency?
ANS A contingency is a proposition that is neither a tautology nor a
contradiction — it is true for some assignments and false for others.
Q5 Define logical conjunction (AND).
ANS The conjunction of p and q, written p ∧ q, is true only when both p and
q are true.
Q6 Define logical disjunction (OR).
ANS The disjunction of p and q, written p ∨ q, is true when at least one of p
or q is true.
,Q7 Define logical negation (NOT).
ANS The negation of p, written ¬p, is true when p is false, and false when p
is true.
Q8 What is the conditional (implication) p → q?
ANS The conditional p → q is false only when p is true and q is false; it is
true in all other cases.
Q9 What is the biconditional p ↔ q?
ANS The biconditional p ↔ q is true when p and q have the same truth value
(both true or both false).
Q10 What is the contrapositive of p → q?
ANS The contrapositive is ¬q → ¬p. It is logically equivalent to the original
conditional.
Q11 What is the converse of p → q?
ANS The converse is q → p. It is NOT logically equivalent to the original
conditional.
Q12 What is the inverse of p → q?
ANS The inverse is ¬p → ¬q. It is logically equivalent to the converse but
NOT to the original conditional.
Q13 What does it mean for two propositions to be logically equivalent?
ANS Two propositions are logically equivalent if they have the same truth
value for every possible assignment of truth values to their variables,
written P ≡ Q.
Q14 State De Morgan's first law.
,ANS ¬(p ∧ q) ≡ ¬p ∨ ¬q. The negation of a conjunction equals the
disjunction of the negations.
Q15 State De Morgan's second law.
ANS ¬(p ∨ q) ≡ ¬p ∧ ¬q. The negation of a disjunction equals the
conjunction of the negations.
Q16 What is exclusive or (XOR)?
ANS The exclusive or of p and q, written p ⊕ q, is true when exactly one of
p or q is true, but not both.
Q17 What is modus ponens?
ANS Modus ponens: Given p → q and p are true, we conclude q is true. It is
a fundamental rule of inference.
Q18 What is modus tollens?
ANS Modus tollens: Given p → q and ¬q are true, we conclude ¬p is true.
Q19 What is the absorption law for OR?
ANS p ∨ (p ∧ q) ≡ p. Absorbing a conjunction into a disjunction leaves p
unchanged.
Q20 What is the distributive law of AND over OR?
ANS p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).
2. Predicates and Quantifiers
Q21 What is a predicate?
ANS A predicate is a function that takes one or more variables and returns a
proposition. Example: P(x) = 'x is even.'
, Q22 What is the universal quantifier?
ANS The universal quantifier ∀ means 'for all.' ∀x P(x) asserts that P(x) is
true for every element x in the domain.
Q23 What is the existential quantifier?
ANS The existential quantifier ∃ means 'there exists.' ∃x P(x) asserts that
P(x) is true for at least one element x in the domain.
Q24 What is the negation of ∀x P(x)?
ANS ¬(∀x P(x)) ≡ ∃x ¬P(x). There exists at least one x for which P(x) is
false.
Q25 What is the negation of ∃x P(x)?
ANS ¬(∃x P(x)) ≡ ∀x ¬P(x). P(x) is false for every x in the domain.
Q26 What is a free variable in a predicate?
ANS A free variable is a variable in a predicate that is not bound by any
quantifier.
Q27 What is a bound variable?
ANS A bound variable is a variable that has been quantified (bound by ∀ or
∃) within a logical expression.
Q28 What is the domain (universe of discourse)?
ANS The domain is the set of all possible values that variables in a predicate
can take.
Q29 What is a counterexample in logic?
ANS A counterexample is a specific value of the variable that makes a
universally quantified statement false.
