Using a proof by contrapositive, prove:
Theorem: For every pair of integers x and y, if x + y is odd, then x is odd or y is odd.
Proof:
Assume that both x and y are even integers. By definition, an even integer can be expressed as 2k, where k is an integer. Therefore, we can write x = 2a and y = 2b, where a and b are integers.
Now, let's consider the sum of x and y:
x + y = 2a + 2b = 2(a + b).
Since a + b is an integer, we can rewrite the sum as:
x + y = 2c, where c = a + b.
From this equation, we can see that x + y is even, not odd. This contradicts our assumption that x + y is odd.
Therefore, our assumption that both x and y are even integers must be false. At least one of them must be odd.
Hence, we have proven that for every pair of integers x and y, if x + y is odd, then x is odd or y is odd.