Tag Info

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:

https://en.wikipedia.org/wiki/Partition_of_a_set

https://proofwiki.org/wiki/Definition:Set_Partition