3. Let r(x) be r is one of the five roommates listed, let d(x) be x has taken a course in discrete mathematics, and let a(x) be x can take a course in algorithms.

Given these premises x(r(x) d(x)) and x(d(x) a(x)), apply rules of inference to conclude x(r(x) a(x)). Let y represent an arbitrary person.

Proofs by Induction

4. Given two odd integers, using direct proof, show that their product is odd.

5. Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contraposition of this theorem.