I need to either prove or disprove that for a set of $n*n$ square matrices where n is some fixed positive number and all entries in the matrices are elements of the real numbers, that an equivalence relation exists such that matrices X and Y are related if the trace of the matrices $Tr(X - Y)$ is an element of the the integers.
I know that to prove an equivalence relation I need to prove reflexivity, symmetry and transitivity of the relation, and so far I have:
Reflexivity: $Tr(X-X) = 0$ and 0 is an element of the integers
Symmetry: Assuming $Tr(X-Y)$ is an element of the integers, then $Tr(Y-X) = -Tr(X-Y)$ and since $Tr(X-Y)$ is an element of the integers, its negation must also be an integer since integers are closed under multiplication.
But I'm getting stuck at transitivity. I'm not sure where to even start with it so any help would be appreciated.
If $\operatorname{tr}(X-Y)\in\Bbb Z$ and $\operatorname{tr}(Y-Z)\in\Bbb Z$, then$$\operatorname{tr}(X-Z)=\operatorname{tr}(X-Y)+\operatorname{tr}(Y-Z)\in\Bbb Z.$$