# Quotient Set Determined by Relation Induced by Partition is That Partition

## Theorem

Let $S$ be a set.

Let $\PP$ be a partition of $S$.

Let $\RR$ be the equivalence relation induced by $\PP$.

Then the quotient set $S / \RR$ of $S$ is $\PP$ itself.

## Proof

Let $P \subseteq S$ such that $P \in \PP$.

Let $x \in P$.

Then:

 $\ds y$ $\in$ $\ds \eqclass x \RR$ $\ds \leadstoandfrom \ \$ $\ds \tuple {x, y}$ $\in$ $\ds \RR$ Definition of Equivalence Class $\ds \leadstoandfrom \ \$ $\ds y$ $\in$ $\ds P$ Definition of Equivalence Relation Induced by Partition

Therefore:

$P = \eqclass x \RR$

and so:

$P \in S / \RR$

and so:

$\PP \subseteq S / \RR$

Now let $x \in S$.

As $\PP$ is a partition:

$\exists P \in \PP: x \in P$

Then by definition of $\RR$:

$\tuple {x, y} \in \RR \iff y \in \eqclass x \RR$
 $\ds y$ $\in$ $\ds P$ $\ds \leadstoandfrom \ \$ $\ds \tuple {x, y}$ $\in$ $\ds \RR$ Definition of Equivalence Relation Induced by Partition $\ds \leadstoandfrom \ \$ $\ds y$ $\in$ $\ds \eqclass x \RR$ Definition of Equivalence Class

Therefore:

$\eqclass x \RR = P$

and so:

$\eqclass x \RR \in \PP$

That is:

$\SS / \RR \subseteq \PP$

It follows by definition of set equality that:

$\SS / \RR = P$

Hence the result.

$\blacksquare$