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

1. A{(f, g) | f(x) − g(x) = 1 ∀x ∈ Z}
2. B{(f, g) | f(0) = g(0) or f(1) = g(1)}
3. C{(f, g) | f(0) = g(1) and f(1) = g(0)}
4. D{(f, g) | f(x) − g(x) = k for some k ∈ Z}

Pick an option to check yourself, or open the answer below.

Show answer & explanation

Correct answer: D. {(f, g) | f(x) − g(x) = k for some k ∈ Z}

Explanation

Differing by a constant is reflexive (k = 0), symmetric (−k) and transitive (k₁ + k₂). (A) isn't reflexive, (B) isn't transitive and (C) isn't reflexive.