"if x and y are odd integers, then x + y is even"

give a proof by contradiction of this theorem.

Hello!

The statement is "if x and y are odd integers, then x + y is even"

and we want to prove it by contradiction.

Suppose that we have x and y odd numbers, and suppose that his addition is odd.

We know that an odd number can be writen as (2n +1) (and a even number can be written as 2n) where n is an integer number; then:

x = (2k + 1) and y = (2m + 1)

and x + y = j, where j is also a odd number, then j = (2h + 1)

then:

2k + 1 + 2m + 1 = 2h + 1

2(k + m) + 2 = 2h + 1

2(k + m) +1 = 2h

if k and m are integers, then k + m is also an integer, suppose that k + m = g

then 2g + 1 = 2h

this says that in odd number is equal to an even number, then we have a contradiction, and the addition of two odd numbers cant be an odd number.