Question

Do not use the same proof approach more than once (use direct, contrapositive, or contradiction). Justify your choice of proof approach for each theory.

A. If n is an even integer, then −3n − 1 is an odd integer.
B. If 5x − 7 is odd, then 9x + 2 is even
C. 11x − 7 is even if and only if x is odd.

Answer 1

To prove Theory A, use direct proof. To prove Theory B, use contradiction. To prove Theory C, use contrapositive.

Step-by-step explanation:

To prove each theory, we will use a different proof approach:

Theory A: If n is an even integer, then −3n − 1 is an odd integer. We will use direct proof. We can assume that n is an even integer, and then show that −3n − 1 is an odd integer.

Theory B: If 5x − 7 is odd, then 9x + 2 is even. We will use contradiction. We can assume that 5x − 7 is odd and 9x + 2 is odd, and then show that this leads to a contradiction.

Theory C: 11x − 7 is even if and only if x is odd. We will use the contrapositive. We can assume that x is not odd, and then show that 11x − 7 is not even.