So I am currently learning about relations and functions in a more formal way than what I am used to. Therefore I decided to tackle some beginning problems, but I have stumbled upon this problem, where I am not quite sure how to solve it:
For a relation $$a \mathcal R b \Leftrightarrow 3|(a^2 - b^2)$$ where $a$ and $b$ are whole numbers, show that this relation is an equivalence relation and determine its partition.
It was quite easy to show that it is indeed an equivalence relation and I also know that the partition of equivalence relations is the set of all their equivalence classes. However, I am not quite sure whether that really answers the problem. So I tried giving a more formal answer like this:
For a given element b of the set of whole numbers the equivalence class contains all integer values of $a = \pm \sqrt{b^2 + 3k}$ where $k$ is just an integer. This comes from $$3|(a^2 - b^2) \Rightarrow a^2 = b^2 + 3k.$$ And the set of all these sets is then the partition. Would this be "determining" the partition or am I misunderstanding something?