The question is, "Which of these relations on the set of all functions from $\mathbb{Z}$ to $\mathbb{Z}$ are equivalence relations.


I just want to make certain that I am interpreting this properly. So, $(f,g)$ is an element in the relation, right? But it can only be an element if $f$ and $g$, evaluated at one, are equal? And when the question says "of all functions," it means functions like $f(x)=x^2$? With this information, it can be reflexive, symmetric, and transitive, only if $f$ and $g$ are the exact same functions, is that correct?


Maybe you're getting to complicated. $f$ equivalent to $f$ means simply $f(1)=f(1)$. And symmetry is: If $f$ equivalent to $g$ then $g$ equivalent to $f$, so this one is simply: If $f(1)=g(1)$ then $g(1)=f(1)$. Transitivity can be done, same idea.

Note that these statements only involve what $f$ is at $x=1$; the value of $f(2)$ for example is irrelevant.


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