Definition:Equivalence Relation Induced by Partition

Definition
Let $$S$$ be a set.

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

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

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

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


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