# Describing the Partition for a given equivalence relation.

In $\mathbb{R}$ let $a\sim b$ iff $a-b$ exists $\in$ $\mathbb{Q}$.

I have already proven that $\sim$ is an equivalence relation.

However, the second part of the question asks to describe the partition associated with the equivalence relation, and this is where I am confused.

How would you describe the partition with this equivalence relation?

$$P = \bigcup_{x\in \Bbb R} \{ \{x+y | y \in \Bbb Q \} \}$$
There are infinitely many sets in the partition; in fact, uncountably many. Each set in the partition is a copy of $\mathbb{Q}$. The union of these copies of $\mathbb{Q}$ make up the whole real line $\mathbb{R}$.