Q1. Consider the following statements: P: "If the alarm rings, then the security guard will arrive." Q: "The security guard has arrived." R: "The alarm did not ring." Determine the truth values of P, Q, and R under different scenarios. Additionally, evaluate the truth value of the compound statement (PR)(QP).

Q2. Consider the proposition Q(n) defined for positive integers n with the following properties: Q(n): "The sum of the first n even positive integers is equal to n(n+1)." Use Strong Induction to prove that Q(n) is true for all positive integers n. Follow the steps of Strong Induction: Step 1 (Base Step): Prove that Q(1) is true. Step 2 (Inductive Step): Prove that for any positive integer k, if Q(1) through Q(k) are all true, then Q(k+1) is true. Specifically, show that the sum of the first k+1 even positive integers equals (k+1)(k+2), assuming that the sum of the first k even positive integers equals k(k+1). You may use the fact that the sum of the first n even positive integers is n(n+1) to assist in your proof.

Q3. Prove the following conjecture by contradiction: "For any positive integer n, if n2 + 3 is odd, then n is even."

Q4. Given the sets C={2,3,6,7} and D={3,7}: a. Draw a Venn diagram of C and D using the universal set U={0,1,2,3,4,5,6,7,8,9} b. Graph C on the number line.