Relations, Equivalence Relations, Partitions

Question

Consider A=$\mathbb{Z}$. For each integer $n$, define

$$B_n = \{{m\in \mathbb{Z}\ | \ (\exists q)(m=n+5q)}\}.$$

Prove that $\{{B_n}\}_{n\in\mathbb{Z}}$ is a partition of $\mathbb{Z}$. Identify the equivalence classes.

Answer 1

$B_n$ is the set of all the numbers that is congruent to $n$ modulo $5$. Now obviously

1. Reflexive: $a\equiv a~(mod~5)$,
2. Transitive: $a\equiv b~(mod~5)$ and $b\equiv c~(mod~5)$ implies $a\equiv c~(mod~5)$,
3. Symmetric: $a\equiv b~(mod~5)$ implies $b\equiv a~(mod~5)$