Show that $P$ is a partition of the corresponding set $A$ and define the equivalence relation induced by partition $P$.

Question

$A = \mathbb N$ and $ P = \{ \{ m \in \mathbb N: m \ \ \text{is a multiple of} \ \ 3\}, \{ m \in \mathbb N: m \ \ \text{is not a multiple of 3}\} \} $.

How can I define the equivalence relation induced by P?

I don't know how to start.

Answer 1

As was noted in the comments, you can simply define the equivalence relation by direct reference to its two equivalence classes:

$$m\sim n\text{ iff }(3\mid m\text{ and }3\mid n)\text{ or }(3\not\mid m\text{ and }3\not\mid n)\;.\tag{1}$$

If you want to compress this into something more elegant, notice that $n^2\equiv 0\pmod 3$ iff $n\mid 3$, and $n^2\equiv 1\pmod3$ iff $3\not\mid n$, so that the relation $(1)$ can also be defined by

$$m\sim n\text{ iff }m^2\equiv n^2\pmod 3$$

or, equivalently,

$$m\sim n\text{ iff }3\mid m^2-n^2\;.$$