1. Which of the following is a valid logical equivalence?
A) ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg
q¬(p∧q)≡¬p∨¬q
B) ¬(p∨q)≡¬p∨¬q\neg(p \lor q) \equiv \neg p \lor \neg
q¬(p∨q)≡¬p∨¬q
C) p∧(q∨r)≡(p∧q)∨(p∧r)p \land (q \lor r) \equiv (p \land q) \lor (p
\land r)p∧(q∨r)≡(p∧q)∨(p∧r)
D) ¬(p→q)≡p∧¬q\neg(p \rightarrow q) \equiv p \land \neg
q¬(p→q)≡p∧¬q
Answer: A) ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg
q¬(p∧q)≡¬p∨¬q
Rationale: This is De Morgan's law for negation of a conjunction,
which states that the negation of a conjunction is the disjunction of the
negations.
2. Which of the following is the correct truth table for the
expression p→(q→r)p \rightarrow (q \rightarrow r)p→(q→r)?
A) T, T, T, T
B) T, F, F, T
C) T, F, T, T
D) T, T, F, F
Answer: C) T, F, T, T
Rationale: The expression p→(q→r)p \rightarrow (q \rightarrow
r)p→(q→r) is true when ppp is true and q→rq \rightarrow rq→r is
,true. The truth table checks this condition, producing the correct
result.
3. What is the truth value of the expression (p∧q)→(r∨s)(p \land
q) \rightarrow (r \lor s)(p∧q)→(r∨s) when
p=True,q=False,r=True,s=Falsep = \text{True}, q =
\text{False}, r = \text{True}, s =
\text{False}p=True,q=False,r=True,s=False?
A) True
B) False
C) Undefined
D) Cannot be determined
Answer: A) True
Rationale: Since p∧qp \land qp∧q is False (because q=Falseq =
\text{False}q=False), the implication (p∧q)→(r∨s)(p \land q)
\rightarrow (r \lor s)(p∧q)→(r∨s) is always True, regardless of the
truth values of rrr and sss.
4. 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).
5. Which of the following represents the contrapositive of the
statement "If ppp, then qqq"?
A) If ¬q\neg q¬q, then ¬p\neg p¬p
B) If qqq, then ppp
C) If ppp, then ¬q\neg q¬q
D) If ¬p\neg p¬p, then ¬q\neg q¬q
Answer: A) If ¬q\neg q¬q, then ¬p\neg p¬p
Rationale: The contrapositive of the statement p→qp \rightarrow
qp→q is ¬q→¬p\neg q \rightarrow \neg p¬q→¬p, which has the
same truth value as the original statement.
6. 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
A) ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg
q¬(p∧q)≡¬p∨¬q
B) ¬(p∨q)≡¬p∨¬q\neg(p \lor q) \equiv \neg p \lor \neg
q¬(p∨q)≡¬p∨¬q
C) p∧(q∨r)≡(p∧q)∨(p∧r)p \land (q \lor r) \equiv (p \land q) \lor (p
\land r)p∧(q∨r)≡(p∧q)∨(p∧r)
D) ¬(p→q)≡p∧¬q\neg(p \rightarrow q) \equiv p \land \neg
q¬(p→q)≡p∧¬q
Answer: A) ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg
q¬(p∧q)≡¬p∨¬q
Rationale: This is De Morgan's law for negation of a conjunction,
which states that the negation of a conjunction is the disjunction of the
negations.
2. Which of the following is the correct truth table for the
expression p→(q→r)p \rightarrow (q \rightarrow r)p→(q→r)?
A) T, T, T, T
B) T, F, F, T
C) T, F, T, T
D) T, T, F, F
Answer: C) T, F, T, T
Rationale: The expression p→(q→r)p \rightarrow (q \rightarrow
r)p→(q→r) is true when ppp is true and q→rq \rightarrow rq→r is
,true. The truth table checks this condition, producing the correct
result.
3. What is the truth value of the expression (p∧q)→(r∨s)(p \land
q) \rightarrow (r \lor s)(p∧q)→(r∨s) when
p=True,q=False,r=True,s=Falsep = \text{True}, q =
\text{False}, r = \text{True}, s =
\text{False}p=True,q=False,r=True,s=False?
A) True
B) False
C) Undefined
D) Cannot be determined
Answer: A) True
Rationale: Since p∧qp \land qp∧q is False (because q=Falseq =
\text{False}q=False), the implication (p∧q)→(r∨s)(p \land q)
\rightarrow (r \lor s)(p∧q)→(r∨s) is always True, regardless of the
truth values of rrr and sss.
4. 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).
5. Which of the following represents the contrapositive of the
statement "If ppp, then qqq"?
A) If ¬q\neg q¬q, then ¬p\neg p¬p
B) If qqq, then ppp
C) If ppp, then ¬q\neg q¬q
D) If ¬p\neg p¬p, then ¬q\neg q¬q
Answer: A) If ¬q\neg q¬q, then ¬p\neg p¬p
Rationale: The contrapositive of the statement p→qp \rightarrow
qp→q is ¬q→¬p\neg q \rightarrow \neg p¬q→¬p, which has the
same truth value as the original statement.
6. 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
