Equivalence classes for a relation between integers

Question

I have the following relation in $\mathbb Z$: $x \sim y$ iff $x-y$ is a multiple of $4$. How do I approach this?

Answer 1

By the division algorithm, for every integer $x$ you can find integers $q$ and $r$ such that $$x=4q+r$$ Moreover, $0\leq r<4$. Therefore, you have at most four equivalence classes $$[0],[1],[2],[3]$$ Moreover, these four equivalence classes are all distinct, because if $0\leq i<j<4$ are integers and $[i]=[j]$, then $4$ must divide $j-i$ and since $j-i<4$ you get that $j-i=0$, hence $i=j$.

Another equivalent approach is the following. Given a group $G$ and a subgroup $H$ of $G$, say that two elements $a, b$ of $G$ are in the relation $R$ if and only if $ab^{-1}\in H$. Then you can check that $R$ is an equivalence relation and the quotient set coincide with the set of right coset of $H$ in $G$. In your case, $G$ is the group of integers with addition and $H$ the subgroup $4\mathbb{Z}$ of integer multiples of $4$, hence the set of all equivalence classes is the factor group $\mathbb{Z}/4\mathbb{Z}$.