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

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?

• There is a natural equivalence relation associated to a partition: $x\sim y$ iff $x$ and $y$ are contained in the same partition set. Jun 8, 2020 at 13:20

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 $$X/E$$.
• Note that the equivalence classes of $E$ are precisely the elements of $P$. So $X/P$ is just $P$. Jun 8, 2020 at 13:29
• @MarkKamsma Is this true that type of $X/P$ is a relation but type of$X/E$ is a partition of X? Jun 8, 2020 at 16:36
• @Amir Both are notation for the same thing. But the type of $P$ is a partition, while the type of $E$ is a relation. Jun 8, 2020 at 16:58