Question

Show that the relation R = {(x,y)lx-y is an even integer} is an equivalence relation on the set of real numbers.

Answer 1

1) If xRy then x-y=2n where n is an integer, and this means y-x=-2n which is also an integer. So yRx implying (y,x) belongs to R. So R is symmetric.

2) xRx and yRy as x-x=0 and y-y=0. These are also even and so R is reflexive.

3) If xRy and yRz then x-y=2n, y-z=2m, and x-z = 2(n+m). This is also even. So xRz implies R is transitive.

So, R is equivalence relation.