Problem 1: Let P, Q, and R be statements. Write down a truth table for the statement ??[ (P or Q) ? (Q and R) ] and (not R)??

Problem 2: Which of the following conditions are necessary for the positive integer n to be divisible by 6? Which are sufficient? (Proofs are not necessary here.)

(1) 3 divides n.

(2) 9 divides n.

(3) 12 divides n.

(4) n = 12.

(5) 6 divides n2.

(6) 2 divides n and 3 divides n.

(7) 2 divides n or 3 divides n.

Problem 3: A tautology is a statement which is always true. Let P and Q be statements.Which of the following four statements are tautologies? Prove that your answer is correct.

(1) ??[ (P ? Q) and P ] ? Q.??

(2) ??[ (P ? Q) and Q ] ? P.??

(3) ??[ (P ? Q) and (not P) ] ? (not Q).??

(4) ??[ (P ? Q) and (not Q) ] ? (not P).??

Problem 4: Let n be an integer. Prove that 0 divides n if and only if n = 0.