Partitions and Equivalence Classes

Let A denote a set. A partition of A is a family of non-empty subsets of A such that each element of A belongs to exactly one member of the family. In other words, a partition of a set A is a collection of non-overlapping, non-empty subsets of A whose union is the set A.

If $\Re$ is an equivalence relation on the set A, then the set of all elements in A equivalent to a given element x₀ is called an equivalence class

A₀ = { a $\in$ A $\vert$ a $\cong x_0$ }