Is the relation $xPy$ iff $ y = x + n\pi$ an equivalence relation on $\Bbb R$?

Question

I am trying to prove that the relation $P$ on $\mathbb{R}$ given by the rule

$$\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$$

From what I can see, $P$ fails the reflexive test, i.e. when showing xPx: $x \neq x + n\pi$.

But I was told by somebody that $P$ is an equivalence relation.

Could someone please confirm whether it is indeed an equivalence relation or not?

Thanks heaps C :)

Answer 1

As you realized in the comments, your issue was in the quantifier: we only need some integer $n$ such that $x = x +n \pi$ for reflexivity (not all $n$); therefore the choice of $n=0$ ensures reflexivity for you.

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Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.