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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?

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

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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 $X/E$.

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

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