Review and Practice 2

1. Direct Proof (from ADS)

Prove that the sum of two odd positive integers is an even positive integer.

2. Proof

Prove the following statement:

If \(n^2\) is odd, then \(n\) is odd.

(Hint: Try a proof by contrapositive or proof by contradiction.)

3. Proof (from ADS)

Prove that if \(x\) and \(y\) are real numbers such that \(x +y \leq 1\), then \(x \leq 1/2\) or \(y \leq 1/2\).

(Hint: Try a proof by contrapositive or proof by contradiction.)

4. Exhaustive Proofs

For finite sets \(X\) and \(Y\), how you would exhaustively prove that \(f: X \to Y\) is a bijection?

5. Proofs over Sets

Prove that if \(x \in A \cap (B \cup C)\), then \(x \in (A \cap B) \cup (A \cap C)\).

(Hint: Translate this to propositional logic using the definitions of intersection and union.)