Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 2 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.