Question

By direct and contra positivity prove that if x is an integer then x^2 - x is even

Answer 1

DIrect proof : if x is an integer, then x is even or x is odd.
case 1: Assume x is even. then x^2 is even , and x is even. and it is closed under subtraction
case 2.Assume x is odd , then x^2 is odd, and x^2 - x is odd - odd = even

Answer 2

ok case 1: assume x is even, then x = 2k for some integer k.
x^2 - x = (2k)^2 - (2k) = 4k^2 - 2k = 2 ( 2k^2 - k) , so x^2 - x = 2m for some m integer, so x^2 - x is even