Which of these relations on the set of all functions from ℤ to ℤ are equivalence relations? Determine the properties of an equivalence relation that the others lack. 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 ∈ ℤ }

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A: (a) For every m ∈ Z, 5 | (m2 −m2) = 0      Hence, m ~ m for all m ∈ Z      Hence, ~ is reflexive.

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A: We are allowed to do a maximum three sub parts as per bartleby guidelines

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Q: Prove that these relations on the set of all functions from Z to Z are equiva- lence relations.…

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Q: d) ((f g)| for some CEZ, for all xE Z, f(x) – g(x) = C} e) {f, 8) | f(0) = g(1) and f(1) = g(0)}

A: Part (d) From the given information: The relation is . Here, C is constant.

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