Question

Problem 2. Which of the following binary relation on the set of all functions from Z to Z are equivalence relations? Prove your answers. (a) {(f,g) : f(1) = g(1)}.

(b) {(f, g) : f(0) = g(0) or f(1) = g(1)}. (c) {(f,g) : f(x) – g(x) = 1 for all x E Z}. (d) {(f,g) : for some c e Z, for all x € Z, f(x) – g(x) = c} (e) {(f,g) : f(0) = g(1) and f(1) = g(0)}.

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