1. Which of the following logical operations is equivalent to
p∨(p∧q)p \lor (p \land q)p∨(p∧q)?
A. ppp
B. qqq
C. ¬p∧q\neg p \land q¬p∧q
D. p∧qp \land qp∧q
Answer: A) ppp
Rationale: The expression simplifies to ppp, because if ppp is true,
the disjunction is true regardless of the value of qqq.
2. Which of the following is an example of 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.
,3. 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.
4. 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.
, 5. 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
D. ¬p→p\neg p \rightarrow p¬p→p
Answer: A) p→pp \rightarrow pp→p
Rationale: A tautology is a statement that is always true. p→pp
\rightarrow pp→p is always true because any proposition implies
itself.
6. Which of the following is the converse 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 qqq
D. If ¬p\neg p¬p, then ¬q\neg q¬q
Answer: B) If qqq, then ppp
Rationale: The converse of p→qp \rightarrow qp→q is q→pq
\rightarrow pq→p, which flips the order of the implication.
7. Which of the following is the correct truth table for the
expression p↔qp \leftrightarrow qp↔q?
p∨(p∧q)p \lor (p \land q)p∨(p∧q)?
A. ppp
B. qqq
C. ¬p∧q\neg p \land q¬p∧q
D. p∧qp \land qp∧q
Answer: A) ppp
Rationale: The expression simplifies to ppp, because if ppp is true,
the disjunction is true regardless of the value of qqq.
2. Which of the following is an example of 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.
,3. 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.
4. 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.
, 5. 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
D. ¬p→p\neg p \rightarrow p¬p→p
Answer: A) p→pp \rightarrow pp→p
Rationale: A tautology is a statement that is always true. p→pp
\rightarrow pp→p is always true because any proposition implies
itself.
6. Which of the following is the converse 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 qqq
D. If ¬p\neg p¬p, then ¬q\neg q¬q
Answer: B) If qqq, then ppp
Rationale: The converse of p→qp \rightarrow qp→q is q→pq
\rightarrow pq→p, which flips the order of the implication.
7. Which of the following is the correct truth table for the
expression p↔qp \leftrightarrow qp↔q?
