Definition:Set Partition/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

Let $S$ be a set.


A partition of $S$ is a set of subsets $\Bbb S$ of $S$ such that:

$(1): \quad$ $\Bbb S$ is pairwise disjoint: $\forall S_1, S_2 \in \Bbb S: S_1 \cap S_2 = \O$ when $S_1 \ne S_2$
$(2): \quad$ The union of $\Bbb S$ forms the whole set $S$: $\ds \bigcup \Bbb S = S$
$(3): \quad$ None of the elements of $\Bbb S$ is empty: $\forall T \in \Bbb S: T \ne \O$.


Also defined as

Some sources do not impose the condition that all sets in $\Bbb S$ are non-empty.

This is most probably more likely to be an accidental omission rather than a deliberate attempt to allow $\O$ to be an element of a partition.

The point is minor; proofs of partitionhood usually include a demonstration that all elements of such a partition are indeed non-empty.


Also known as

A partition is also known a decomposition.

This same definition is also encountered in the field of combinatorics.

Some sources use the word dissection, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ that term is reserved for the geometric context.


Also see

  • Results about set partitions can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Set_Partition/Definition_1&oldid=708536"