What is meaning of $X/P$? ($X$ is a set and $P$ is a partition)

Question

The definition of $x/E$ when $E$ is an equivalence relation is : $$x/E = \{y\in X \mid (y,x)\in E \},$$ and the definition of $X/E$: $$X/E = \{x/E\ \mid x\in X\}.$$ Now, what is $X/P$ when $P$ is a non-empty partition of X?

Answer 1

A partition $P$ defines uniquely an equivalence relation $E$ by setting up

$$ (x,y) \in E \iff \exists p \in P (x \in p \wedge y \in p)$$

Then $X/P$ is just $X/E$.