1. 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.
2. What is the logical negation of the statement "There exists an
xxx such that P(x)P(x)P(x) is true"?
A. For all xxx, P(x)P(x)P(x) is false.
B. For some xxx, P(x)P(x)P(x) is false.
C. There exists an xxx such that P(x)P(x)P(x) is false.
D. There exists an xxx such that P(x)P(x)P(x) is true.
Answer: A) For all xxx, P(x)P(x)P(x) is false.
,Rationale: The negation of the existential quantifier ∃xP(x)\exists
x P(x)∃xP(x) becomes a universal quantifier ∀x¬P(x)\forall x
\neg P(x)∀x¬P(x).
3. 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=Truep = \text{True}p=True and q=Falseq =
\text{False}q=False?
A. True
B. False
C. Undefined
D. Cannot be determined
Answer: A) True
Rationale: When p=Truep = \text{True}p=True and q=Falseq =
\text{False}q=False, ¬(p∨q)\neg(p \lor q)¬(p∨q) is true, and
p∧¬qp \land \neg qp∧¬q is also true, making the implication true.
4. Which of the following is the negation of the statement
"∀x∈A,∃y∈B,P(x,y)\forall x \in A, \exists y \in B, P(x,
y)∀x∈A,∃y∈B,P(x,y)"?
A. ∃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)
B. ∃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)
, 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, \neg \exists y \in B, P(x,
y)∃x∈A,¬∃y∈B,P(x,y)
Answer: A) ∃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)
Rationale: The negation of the statement involving both universal
and existential quantifiers changes the quantifiers accordingly and
negates the predicate.
5. Which of the following is the negation of the statement "For
every student, there is a course they are enrolled in"?
A. For every student, there is no course they are enrolled in.
B. There exists a student who is not enrolled in any course.
C. There exists a course that no student is enrolled in.
D. For some students, there is no course they are enrolled in.
Answer: B) There exists a student who is not enrolled in any
course.
Rationale: The negation of a universal quantifier statement
∀x∃yP(x,y)\forall x \exists y P(x, y)∀x∃yP(x,y) becomes
∃x¬∃yP(x,y)\exists x \neg \exists y P(x, y)∃x¬∃yP(x,y), which
means there exists a student not enrolled in any course.
(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.
2. What is the logical negation of the statement "There exists an
xxx such that P(x)P(x)P(x) is true"?
A. For all xxx, P(x)P(x)P(x) is false.
B. For some xxx, P(x)P(x)P(x) is false.
C. There exists an xxx such that P(x)P(x)P(x) is false.
D. There exists an xxx such that P(x)P(x)P(x) is true.
Answer: A) For all xxx, P(x)P(x)P(x) is false.
,Rationale: The negation of the existential quantifier ∃xP(x)\exists
x P(x)∃xP(x) becomes a universal quantifier ∀x¬P(x)\forall x
\neg P(x)∀x¬P(x).
3. 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=Truep = \text{True}p=True and q=Falseq =
\text{False}q=False?
A. True
B. False
C. Undefined
D. Cannot be determined
Answer: A) True
Rationale: When p=Truep = \text{True}p=True and q=Falseq =
\text{False}q=False, ¬(p∨q)\neg(p \lor q)¬(p∨q) is true, and
p∧¬qp \land \neg qp∧¬q is also true, making the implication true.
4. Which of the following is the negation of the statement
"∀x∈A,∃y∈B,P(x,y)\forall x \in A, \exists y \in B, P(x,
y)∀x∈A,∃y∈B,P(x,y)"?
A. ∃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)
B. ∃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)
, 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, \neg \exists y \in B, P(x,
y)∃x∈A,¬∃y∈B,P(x,y)
Answer: A) ∃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)
Rationale: The negation of the statement involving both universal
and existential quantifiers changes the quantifiers accordingly and
negates the predicate.
5. Which of the following is the negation of the statement "For
every student, there is a course they are enrolled in"?
A. For every student, there is no course they are enrolled in.
B. There exists a student who is not enrolled in any course.
C. There exists a course that no student is enrolled in.
D. For some students, there is no course they are enrolled in.
Answer: B) There exists a student who is not enrolled in any
course.
Rationale: The negation of a universal quantifier statement
∀x∃yP(x,y)\forall x \exists y P(x, y)∀x∃yP(x,y) becomes
∃x¬∃yP(x,y)\exists x \neg \exists y P(x, y)∃x¬∃yP(x,y), which
means there exists a student not enrolled in any course.
