# Category:Set Partitions

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This category contains results about Set Partitions.

Definitions specific to this category can be found in Definitions/Set Partitions.

Let $S$ be a set.

### Definition 1

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 $S$: $\displaystyle \bigcup \Bbb S = S$
- $(3): \quad$ None of the elements of $\Bbb S$ is empty: $\forall T \in \Bbb S: T \ne \O$.

### Definition 2

A **partition** of $S$ is a set of non-empty subsets $\Bbb S$ of $S$ such that each element of $S$ lies in exactly one element of $\Bbb S$.

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### B

### E

### P

### Q

### R

### S

## Pages in category "Set Partitions"

The following 14 pages are in this category, out of 14 total.