let x,y be elements of integers. proove x-y is even if and only if x and y of are same parity.?

Answer 1

You have to do two cases. I will do one of them for you and then the other one will be similar.
Case 1) Suppose x and y are both odd, then
x = 2m+1 for some integer m and
y = 2n + 1 for some integer n
x - y = 2m+1 - (2n+1)
= 2m + 1 - 2n - 1
= 2m - 2n
= 2(m-n)
Thus, x-y is even
Case 2) Suppose x and y are both even.

Answer 2

No need to prove anything, let's just assume the integers are innocent and their parity is a personal choice. Do let x and y be, and please don't say disparaging things about integers you hardly even know.