1. 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.
2. Which of the following is logically equivalent to the expression
(p∧q)∨(p∧¬q)(p \land q) \lor (p \land \neg q)(p∧q)∨(p∧¬q)?
A. ppp
B. qqq
C. p∨qp \lor qp∨q
D. ¬p∨q\neg p \lor q¬p∨q
Answer: A) ppp
Rationale: The expression simplifies to ppp because it is a
disjunction involving two terms where both terms contain ppp.
,3. Which of the following is logically equivalent to the expression
¬(p→q)\neg(p \rightarrow q)¬(p→q)?
A. p∧¬qp \land \neg qp∧¬q
B. p∨¬qp \lor \neg qp∨¬q
C. ¬p∧q\neg p \land q¬p∧q
D. p↔qp \leftrightarrow qp↔q
Answer: A) p∧¬qp \land \neg qp∧¬q
Rationale: The negation of an implication p→qp \rightarrow
qp→q is logically equivalent to p∧¬qp \land \neg qp∧¬q.
4. Which of the following represents the negation of the
statement "∃x∈A,∀y∈B,P(x,y)\exists x \in A, \forall y \in B, P(x,
y)∃x∈A,∀y∈B,P(x,y)"?
A. ∀x∈A,∃y∈B,¬P(x,y)\forall x \in A, \exists y \in B, \neg P(x,
y)∀x∈A,∃y∈B,¬P(x,y)
B. ∃x∈A,∀y∈B,¬P(x,y)\exists x \in A, \forall y \in B, \neg P(x,
y)∃x∈A,∀y∈B,¬P(x,y)
C. ∀x∈A,∃y∈B,¬P(x,y)\forall x \in A, \exists y \in B, \neg P(x,
y)∀x∈A,∃y∈B,¬P(x,y)
D. ∃x∈A,∃y∈B,¬P(x,y)\exists x \in A, \exists y \in B, \neg P(x,
y)∃x∈A,∃y∈B,¬P(x,y)
, Answer: D) ∃x∈A,∃y∈B,¬P(x,y)\exists x \in A, \exists y \in B,
\neg P(x, y)∃x∈A,∃y∈B,¬P(x,y)
Rationale: The negation of an existential and universal quantifier
statement changes the quantifiers and negates the predicate.
5. Which of the following statements is logically equivalent to
p→(q∧r)p \rightarrow (q \land r)p→(q∧r)?
A. p→q∧p→rp \rightarrow q \land p \rightarrow rp→q∧p→r
B. (p→q)∧(p→r)(p \rightarrow q) \land (p \rightarrow
r)(p→q)∧(p→r)
C. p∨(q∧r)p \lor (q \land r)p∨(q∧r)
D. p∧(q∨r)p \land (q \lor r)p∧(q∨r)
Answer: B) (p→q)∧(p→r)(p \rightarrow q) \land (p \rightarrow
r)(p→q)∧(p→r)
Rationale: The implication p→(q∧r)p \rightarrow (q \land
r)p→(q∧r) is equivalent to (p→q)∧(p→r)(p \rightarrow q) \land (p
\rightarrow r)(p→q)∧(p→r), as both implications must hold for
p→(q∧r)p \rightarrow (q \land r)p→(q∧r) to be true.
6. Which of the following is a tautology?
A. p→pp \rightarrow pp→p
B. p∧¬pp \land \neg pp∧¬p
C. p→¬pp \rightarrow \neg pp→¬p
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.
2. Which of the following is logically equivalent to the expression
(p∧q)∨(p∧¬q)(p \land q) \lor (p \land \neg q)(p∧q)∨(p∧¬q)?
A. ppp
B. qqq
C. p∨qp \lor qp∨q
D. ¬p∨q\neg p \lor q¬p∨q
Answer: A) ppp
Rationale: The expression simplifies to ppp because it is a
disjunction involving two terms where both terms contain ppp.
,3. Which of the following is logically equivalent to the expression
¬(p→q)\neg(p \rightarrow q)¬(p→q)?
A. p∧¬qp \land \neg qp∧¬q
B. p∨¬qp \lor \neg qp∨¬q
C. ¬p∧q\neg p \land q¬p∧q
D. p↔qp \leftrightarrow qp↔q
Answer: A) p∧¬qp \land \neg qp∧¬q
Rationale: The negation of an implication p→qp \rightarrow
qp→q is logically equivalent to p∧¬qp \land \neg qp∧¬q.
4. Which of the following represents the negation of the
statement "∃x∈A,∀y∈B,P(x,y)\exists x \in A, \forall y \in B, P(x,
y)∃x∈A,∀y∈B,P(x,y)"?
A. ∀x∈A,∃y∈B,¬P(x,y)\forall x \in A, \exists y \in B, \neg P(x,
y)∀x∈A,∃y∈B,¬P(x,y)
B. ∃x∈A,∀y∈B,¬P(x,y)\exists x \in A, \forall y \in B, \neg P(x,
y)∃x∈A,∀y∈B,¬P(x,y)
C. ∀x∈A,∃y∈B,¬P(x,y)\forall x \in A, \exists y \in B, \neg P(x,
y)∀x∈A,∃y∈B,¬P(x,y)
D. ∃x∈A,∃y∈B,¬P(x,y)\exists x \in A, \exists y \in B, \neg P(x,
y)∃x∈A,∃y∈B,¬P(x,y)
, Answer: D) ∃x∈A,∃y∈B,¬P(x,y)\exists x \in A, \exists y \in B,
\neg P(x, y)∃x∈A,∃y∈B,¬P(x,y)
Rationale: The negation of an existential and universal quantifier
statement changes the quantifiers and negates the predicate.
5. Which of the following statements is logically equivalent to
p→(q∧r)p \rightarrow (q \land r)p→(q∧r)?
A. p→q∧p→rp \rightarrow q \land p \rightarrow rp→q∧p→r
B. (p→q)∧(p→r)(p \rightarrow q) \land (p \rightarrow
r)(p→q)∧(p→r)
C. p∨(q∧r)p \lor (q \land r)p∨(q∧r)
D. p∧(q∨r)p \land (q \lor r)p∧(q∨r)
Answer: B) (p→q)∧(p→r)(p \rightarrow q) \land (p \rightarrow
r)(p→q)∧(p→r)
Rationale: The implication p→(q∧r)p \rightarrow (q \land
r)p→(q∧r) is equivalent to (p→q)∧(p→r)(p \rightarrow q) \land (p
\rightarrow r)(p→q)∧(p→r), as both implications must hold for
p→(q∧r)p \rightarrow (q \land r)p→(q∧r) to be true.
6. Which of the following is a tautology?
A. p→pp \rightarrow pp→p
B. p∧¬pp \land \neg pp∧¬p
C. p→¬pp \rightarrow \neg pp→¬p
