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

$\{(f,g)|f(1)=g(1)\}$

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 if f and g are the exact same functions, is that correct?

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up vote 1 down vote accepted

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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