It looks like there is a typo in the definition of S. It should be
S = {(x,y) : x, y ∈ Z and x - y = 2n for some n ∈ Z}
Note that, with respect to this equivalence relation, all even numbers are equivalent to 0 (since every even number is of the form 2n for some integer n) and all odd numbers are equivalent to 1 (odd numbers are of the form 2n + 1 for n ∈ Z). Since 1 - 0 = 1 is not divisible by 2, 1 is not equivalent to 0. So the equivalence classes are [0] and [1]. Since every integer is even or odd, {[0],[1]} partitions Z.
