Question

Give a direct proof for each of the following statements about the integers.

If x is even and y is odd, then x + y is odd.

Answer 1

Answer:

See explanation below

Step-by-step explanation:

If x is even, then we know that x = 2k for a k ≥1

If y is odd, then we know that y = 2j -1 for a j≥1

Therefore, x + y can be written similarly as:

$x+y$

$2k +(2j-1)\\2(k+j)-1$

And clearly, 2(k + j) -1 is odd.

Therefore, the sum x + y is an odd number.