There is a bijection between ways to partition a set \(A\) and the equivalence relations on \(A\)
Every partition gives rise to a distinct equivalence relation
Define \(a \sim b\) to mean \(a,b\) are in the same part. This is a reflective, symmetric, and transitive relation given the definition of a partition.
Every equivalence relation gives rise to a distinct partition.
Define a subset \(X \subseteq A\) as \(\sim\)-closed if, for every \(x \in X\) and \(x' \sim x\), we have \(x' \in X\).
Define a subset \(X \subseteq A\) as \(\sim\)-connected if it is nonempty and \(\forall x,y \in X:\ x \sim y\)
The parts corresponding to \(\sim\) are precisely the \(\sim\)-closed and \(\sim\)-connected subsets.