# Math 55 Fall 2022 Quiz Questions

## Week 1

### Discussion 101 and 102 (Scott)

#### Problem 1

Let $p$ be the proposition "Scott is in the room.""

Let $q$ be the proposition "Today is Friday."

Let $r$ be the proposition "It is time for section."

**(a)** Translate $\neg r \to (\neg p \lor q)$ into an English sentence.

**Answer:** "If it is not time for section, then Scott is not in the room or today is Friday."

**(b)** Rewrite the proposition

"Scott is in the room if and only if today is not Friday and it is time for section"

in terms of propositional variables $p,q,r$ and logical operators.

**Answer:** $p \leftrightarrow (\neg q \land r)$

#### Problem 2

Construct a truth table for each of these compound propositions:

**(a)** $(p \land q) \oplus (p \lor q)$

**Answer:**
$$
\begin{array}{|c|c|c|c|c|}
\hline
p & q & p \land q & p \lor q & (p \land q) \oplus (p \lor q) \\ \hline
T & T & T & T & F \\ \hline
T & F & F & T & T \\ \hline
F & T & F & T & T \\ \hline
F & F & F & F & F \\ \hline
\end{array}
$$

**(b)** $p \to (\neg(q \leftrightarrow r) \land p)$

**Answer:**
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
p & q & r & q \leftrightarrow r & \neg(q \leftrightarrow r) & \neg(q \leftrightarrow r) \land p & p \to (\neg(q \leftrightarrow r) \land p) \\ \hline
T & T & T & T & F & F & F \\ \hline
T & T & F & F & T & T & T \\ \hline
T & F & T & F & T & T & T \\ \hline
T & F & F & T & F & F & F \\ \hline
F & T & T & T & F & F & T \\ \hline
F & T & F & F & T & F & T \\ \hline
F & F & T & F & T & F & T \\ \hline
F & F & F & T & F & F & T \\ \hline
\end{array}
$$