"Comprehensive Answer Key for WGU D420 Discrete Math Logic"

1. Which of the following statements is logically equivalent to the
negation of the statement "If it rains, then I will go to the store"?
A. If it rains, then I will not go to the store.
B. If I do not go to the store, then it does not rain.
C. It rains, and I will not go to the store.
D. I will not go to the store if it rains.
Answer: C) It rains, and I will not go to the store.

Rationale: The negation of an implication p→qp \rightarrow
qp→q is p∧¬qp \land \neg qp∧¬q, meaning both ppp (it rains)
and ¬q\neg q¬q (I will not go to the store).


2. Which of the following statements is logically equivalent to the
expression (p∧q)∨¬r(p \land q) \lor \neg r(p∧q)∨¬r?

A. (p∨q)∧¬r(p \lor q) \land \neg r(p∨q)∧¬r

B. p∧(q∨¬r)p \land (q \lor \neg r)p∧(q∨¬r)

C. p∧q∨¬rp \land q \lor \neg rp∧q∨¬r

D. (p∨¬r)∧q(p \lor \neg r) \land q(p∨¬r)∧q

Answer: C) p∧q∨¬rp \land q \lor \neg rp∧q∨¬r

Rationale: The expression (p∧q)∨¬r(p \land q) \lor \neg
r(p∧q)∨¬r is already in its simplest form and no further

,simplification is needed. Option C maintains the structure of the
original expression.


3. Which of the following is the result of the logical expression
p∨(q∧¬p)p \lor (q \land \neg p)p∨(q∧¬p)?

A. q∨pq \lor pq∨p
B. ppp

C. q∧¬pq \land \neg pq∧¬p
D. ¬p\neg p¬p

Answer: A) q∨pq \lor pq∨p
Rationale: The expression simplifies by applying distribution:
p∨(q∧¬p)p \lor (q \land \neg p)p∨(q∧¬p) is logically equivalent to
q∨pq \lor pq∨p.


4. What is the truth value of the statement ¬(p∨q)→(p∧¬q)\neg
(p \lor q) \rightarrow (p \land \neg q)¬(p∨q)→(p∧¬q) when
p=False,q=Truep = \text{False}, q =
\text{True}p=False,q=True?
A. True
B. False
C. Undefined
D. Cannot be determined
Answer: B) False

, Rationale: When p=Falsep = \text{False}p=False and q=Trueq =
\text{True}q=True, ¬(p∨q)\neg(p \lor q)¬(p∨q) becomes False.
Since the antecedent is False, the entire implication is False.


5. Which of the following statements is a contradiction?

A. p∧¬pp \land \neg pp∧¬p

B. p∨¬pp \lor \neg pp∨¬p
C. p→pp \rightarrow pp→p

D. p↔pp \leftrightarrow pp↔p

Answer: A) p∧¬pp \land \neg pp∧¬p
Rationale: A contradiction is a statement that is always false.
p∧¬pp \land \neg pp∧¬p is a contradiction because ppp and
¬p\neg p¬p cannot both be true.


6. Which of the following is the correct truth table for the
expression p∧(p→q)p \land (p \rightarrow q)p∧(p→q)?
A. T, T, T, F
B. T, F, F, T
C. T, F, T, F
D. T, F, F, F
Answer: A) T, T, T, F