This tag is for questions relating to "partition of a set" or, "set-partition", which is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in exactly one subset.

**Partition of a Set** or, **Set Partition** is division of a set of objects into a family of subsets that are mutually exclusive and jointly exhaustive; that is, no element of the original set is present in more than one of the subsets, and all the subsets together contain all the members of the original set.

**Definition:**

A partition of $~S~$ is a set of subsets $~\mathbb{S}~$ of $~S~$ such that:

$(1):~$ $~\mathbb{S}~$ is pairwise disjoint: $~~∀~S_1,~S_2~∈~\mathbb{S}~:~S_1∩S_2=\phi~$ when $~S_1≠S_2~$

$(2):~$ The union of $~\mathbb{S}~$ forms the whole set $~S~:~~ \cup~\mathbb{S}~=S~$

$(3):~$ None of the elements of $~\mathbb{S}~$ is empty$~: ~∀~T~∈~\mathbb{S}~:~~T≠\phi~$.

The number of partitions of the set $~\{k\}_{k=1}^n~$ is called a

**Bell number**.Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation.

A set equipped with an equivalence relation or a partition is sometimes called a

**setoid**, typically in type theory and proof theory.

**References:**