Find the equivalence classes

Question

Prove or disprove: There is an equivalence relation $\sim$ on $\mathbb{Z}$ defined by $x \sim y$ if $x − y$ is even. What are the equivalence classes?

I have proven that there is an equivalence relation by proving symmetry, transitivity, and reflexivity. How do I go about partitioning $\mathbb{Z}$ into the equivalence classes?

Answer 1

The equivalent classes are: $[0]$, and $[1]$.

$[0] = \{x: x \in \mathbb{Z}, \text{and is even}\}$

$[1] = \{x: x \in \mathbb{Z}, \text{and is odd}\}$.

And if you define your relation: $x \sim y \iff x \equiv y \pmod n$, then you have $n$ equivalent classes:

$[0], [1], ..., [n-1]$

Answer 2

If $x$ is even and $x\sim y$, then $y$ is even. Conversely if $x$ and $y$ are even, $x\sim y$. So a class is the set of even numbers.
Same reasoning for odd numbers.