Definition:Equivalence Relation Induced by Partition

Definition
Let $S$ be a set.

Let $\Bbb S$ be a partition of a set $S$.

Let $\mathcal R \subseteq S \times S$ be the relation defined as:
 * $\forall \left({x, y}\right) \in S \times S: \left({x, y}\right) \in \mathcal R \iff \exists S \in \Bbb S: \left\{{x, y}\right\} \subseteq S$

It is proved in Equivalence Relation Defined by a Partition that:
 * $\mathcal R$ is unique;
 * $\mathcal R$ is an equivalence relation on $S$;

Hence $\Bbb S$ is the quotient set of $S$ by $\mathcal R$, that is:


 * $\Bbb S = S / \mathcal R$