# Homework 1

Due Monday, 9/18:

1. Write a truth table for the proposition: $(p \vee r) \wedge (q \vee r)$
2. Prove that $(p \wedge q) \vee r \equiv (p \vee r) \wedge (q \vee r)$.
3. 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. $\lnot p \rightarrow \lnot q$
3. $\lnot q \rightarrow \lnot p$
4. Write the converse, inverse, and contrapositive of the following conditional statement: “If the length of the side of a square is 5 inches, then the area of the square is 25 square inches.” Determine which of these conditional statements (the original, converse, inverse, and contrapositive) are true.
5. (This one may take some time to think about, so get started ASAP!) Prove by induction: the sum of the first n positive integers is (n)(n+1)/2.