The question is, "Which of these relations on the set of all functions from $Z$ to $Z$ are equivalence relations."
The first relation to consider is, $\{(f,g)|f(0)=g(0)\vee f(1)=g(1)\}$ For this one, I am not very certain where to begin.
The second one, $\{(f,g)|f(x)-g(x)=1,\text{for all } x \in Z\}$ I can see how this isn't reflexive, because $f(x)-f(x)= 0$ is always true for any function. I can also see how it isn't symmetric, because although $f(x)-g(x)=1$ could be true, it's counter-part, $g(x)-f(x)=-1$, won't be true. Despite me being able to see those facts, I can not see how it isn't transitive. How would I show that?
The third one, $\{(f,g)|\text{for some }C~\in Z,\text{for all } x\in Z, f(x)-g(x)=C\}$ I had the idea that it wasn't an equivalence relation based on the fact that if we let $f(x)= x$, and $g(x) = x -1$, and say $x=1$, then $f(1)-g(1)=C \rightarrow C=1$, but $g(1)-f(1)=C \rightarrow C=-1$. The C values aren't the same, implying the relation wouldn't be symmetric, right? Also, I am not sure how to prove or disprove that the relation is transitive.
The last one is similar to the first one: $\{(f,g)|f(0)=g(1)\wedge f(1)=g(0)\}$; and like the first one, I am not sure where to begin.
Sorry that this post is rather long. But thank you for reading! and I hope that you can help me.