Question

Theorem: For any two integers, x and y, if x + y is even, then x and y have the same parity. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

a. Assumed:either x and y re both odd Proven: x+y is even or x and y are both even.
b. Assumed: either x is odd Proven: x+y is even
c. Assumed: either x and y is even or x is even and y is odd. and y are both odd or x and y are both even.
Proven: x+y is odd
d. Assumed: either x is odd and y is even or x is even and y is odd. Proven: x+y is odd

Answer 1

In a proof by contrapositive of the given theorem, option a assumes that either x and y are both odd, and proves that x + y is even or x and y are both even.

Step-by-step explanation:

The theorem states that if the sum of two integers, x and y, is even, then x and y have the same parity (i.e., they are both even or both odd). In a proof by contrapositive, we assume the negation of the conclusion and prove the negation of the hypothesis. Let's analyze each option:

a. Assumed: either x and y are both odd. Proven: x + y is even or x and y are both even.

b. Assumed: either x is odd. Proven: x + y is even.

c. Assumed: either x and y are both even or x is even and y is odd. Proven: x + y is odd.

d. Assumed: either x is odd and y is even or x is even and y is odd. Proven: x + y is odd.

Therefore, the correct answer is a. The assumptions and proven facts in a proof by contrapositive are logically equivalent to those in the original theorem.