Which of the following is an equivalence relation on the set of all functions from Z to Z ?

The relation $f(x) - g(x) = k$ for some fixed integer k is reflexive (take k = 0), symmetric (replace k by −k) and transitive (add the two constants), so it is an equivalence relation. A difference of exactly 1 is not reflexive, since $f - f = 0$. The relation "$f(0) = g(0)$ or $f(1) = g(1)$" is not transitive, because f may agree with g at 0 and g with h at 1 while f and h agree at neither point, and the crossed condition $f(0) = g(1)$ and $f(1) = g(0)$ is not reflexive, since it would need $f(0) = f(1)$ for every f.