# Homework 1

Due Wednesday, 2/14:

The following exercises should be turned in. Make sure to write neatly, put your name on every page, and staple if you have multiple pages.

1. Write a truth table for the proposition: $(p \wedge r) \vee (q \wedge r)$
2. Prove that $(p \vee q) \wedge r \equiv (p \wedge r) \vee (q \wedge r)$.
3. Prove that $(p \rightarrow q) \wedge (\lnot p \rightarrow q) \equiv q$. Let x be a real number (make no assumptions about x besides the fact that it is real). Let p be the statement “$x \geq 0$” and q the statement “$x^2 \geq 0$“. Explain (in words) why $p \rightarrow q$ and $\lnot p \rightarrow q$ are both true statements. Can you conclude that q is true? Why or why not?
4. Suppose p is “The shape S is a square” and q is “The shape S is a rectangle.” Translate the following statements to English, and determine which of these statements are true:
1. $p \leftrightarrow q$
2. $p \rightarrow q$
3. $\lnot p \rightarrow \lnot q$
5. Write the converse, inverse, and contrapositive of the following conditional statement: “If the length of the side of a square is 10 inches, then the area of the square is 100 square inches.” Determine which of these conditional statements (the original, converse, inverse, and contrapositive) are true.

The following exercises from the textbook are optional and should not be turned in:

Section 1.1: #9, 13, 31, 37