1. 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.
2. Which of the following is the correct truth table for p↔qp
\leftrightarrow qp↔q?
A. T, T, F, F
B. T, F, F, T
C. F, T, T, F
D. T, T, T, T
Answer: A) T, T, F, F
,Rationale: The biconditional p↔qp \leftrightarrow qp↔q is true
when both ppp and qqq are the same, and false when they are
different.
3. 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.
4. Which of the following is logically equivalent to the expression
¬(p∨q)∧(p→q)\neg(p \lor q) \land (p \rightarrow
q)¬(p∨q)∧(p→q)?
A. ¬p∧¬q\neg p \land \neg q¬p∧¬q
B. ¬p∧q\neg p \land q¬p∧q
, C. ¬p∨q\neg p \lor q¬p∨q
D. p→qp \rightarrow qp→q
Answer: A) ¬p∧¬q\neg p \land \neg q¬p∧¬q
Rationale: Applying De Morgan's law to ¬(p∨q)\neg(p \lor
q)¬(p∨q) gives ¬p∧¬q\neg p \land \neg q¬p∧¬q, and the
conjunction with p→qp \rightarrow qp→q simplifies to this
result.
5. 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.
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.
2. Which of the following is the correct truth table for p↔qp
\leftrightarrow qp↔q?
A. T, T, F, F
B. T, F, F, T
C. F, T, T, F
D. T, T, T, T
Answer: A) T, T, F, F
,Rationale: The biconditional p↔qp \leftrightarrow qp↔q is true
when both ppp and qqq are the same, and false when they are
different.
3. 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.
4. Which of the following is logically equivalent to the expression
¬(p∨q)∧(p→q)\neg(p \lor q) \land (p \rightarrow
q)¬(p∨q)∧(p→q)?
A. ¬p∧¬q\neg p \land \neg q¬p∧¬q
B. ¬p∧q\neg p \land q¬p∧q
, C. ¬p∨q\neg p \lor q¬p∨q
D. p→qp \rightarrow qp→q
Answer: A) ¬p∧¬q\neg p \land \neg q¬p∧¬q
Rationale: Applying De Morgan's law to ¬(p∨q)\neg(p \lor
q)¬(p∨q) gives ¬p∧¬q\neg p \land \neg q¬p∧¬q, and the
conjunction with p→qp \rightarrow qp→q simplifies to this
result.
5. 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.
