Checking whether a relation is an equivalence relation or not

Question

I am a bit confused whether the given relation is an equivalence relation or not. $\sim$ is defined on the set of all integers $x \sim y$ if and only if $x - y$ is a multiple of $4$. I think this is an equivalence relation. I checked it for reflexivity, symmetry, and transitivity and according to my scratch work, all of these conditions seem to be met. Am I missing something or am I on the right track? Any input would be helpful. Thanks!

Answer 1

Yes you are right, and for showing that, that is the demonstration

x-y is a multiple of 4 so $x-y=4k$ that mean $4/x-y$ implies that $x=y[4]$

And we know $x=y[4]$$\Leftrightarrow $$y=x[4]$ so this relation is symmetry

and also we know $x=x[4]$ so it's reflexive

And we know if $x=y[4] $ and $y=z[4]$ that's implies$ x=z[4]$ So it's transitive

And after this(transitivity, reflexivity, symmetry) we can say this relation is an equivalence relation