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.

Write a truth table for the proposition: $(p \wedge r) \vee (q \wedge r)$
Prove that $(p \vee q) \wedge r \equiv (p \wedge r) \vee (q \wedge r)$ .
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?
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. $\lnot p \rightarrow \lnot q$
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.