Let $n$ be an integer. On the set $F$ of all integer-valued functions of a set $A$, suppose we define $f$ and $g$ to be related if $f(a)\equiv g(a)\pmod{n}$ for every $a\in A$. Is this an equivalence relation on $F$? Prove that it is, or explain why it isn’t.
I think maybe the integer-valued function is screwing me up. I get what it means; you put a value into a function and get an integer back (ceiling and floor are good examples), but I don't see how it affects the problem. I know I have to prove that it's reflexive, symmetric and transitive (or explain why it isn't), but I can't even figure out a starting point.
Yes, there is an equivalence relation on F.
The relation is reflexive, symmetric, and transitive.
Reflexive: for every $f\in F$ and $a\in A$, $f(a)\equiv f(a)\pmod{n}$. This is true.
Symmetric: for every $f, g\in F$ and $a\in A$, if $f(a)\equiv g(a)\pmod{n}$, then $g(a)\equiv f(a)\pmod{n}$. This is true.
Transitive: for every $f, g, k \in F$, and $a\in A$, where $f(a)\equiv g(a)\pmod{n}$ and $g(a)\equiv k(a)\pmod{n}$, then $f(a)\equiv k(a)\pmod{n}$. This is true.