Correspondence Theorem (Set Theory)

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $\mathscr A$ be the set of partitions of $S$ associated with equivalence relations $\mathcal R'$ on $S$ such that:


 * $\left({x, y}\right) \in \mathcal R \implies \left({x, y}\right) \in \mathcal R'$

Then there exists a bijection $\phi$ from $\mathscr A$ onto the set of partitions of the quotient set $S / \mathcal R$.