Relation Induced by Partition is Equivalence

Theorem
Let $$\mathbb S$$ be a partition of a set $$S$$.

Then there exists a unique equivalence relation $$\mathcal{R}$$ on $$S$$ for which $$\mathbb S$$ is the quotient of $$S$$ by $$\mathcal{R}$$, that is:


 * $$\mathbb S = S / \mathcal{R}$$

Proof
If $$\mathbb S$$ partitions $$S$$, we can define the relation $$\mathcal{R}$$ on $$S$$ by:


 * $$\mathcal{R} = \left\{{\left({x, y}\right): \left({\exists T \in \mathbb S: x \in T \land y \in T}\right)}\right\}$$

We now need to show that $$\mathcal{R}$$ is an equivalence relation.

Checking in turn each of the critera for equivalence:

Reflexive
$$\mathcal{R}$$ is reflexive:

$$ $$

Symmetric
$$\mathcal{R}$$ is symmetric:

$$ $$ $$ $$

Transitive
$$\mathcal{R}$$ is transitive:

$$ $$ $$

So $$\mathcal{R}$$ is reflexive, symmetric and transitive, therefore $$\mathcal{R}$$ is an equivalence relation.


 * Now by definition of a partition, we have that:


 * $$\mathbb S$$ partitions $$S \implies \forall x \in S: \exists T \in \mathbb S: x \in T$$

Also:

$$ $$

Thus $$\mathbb S$$ is the family of $$\mathcal{R}$$-classes constructed above.