# Definition:Pairwise Disjoint

From ProofWiki

## Definition

### Set of Sets

A set of sets $\Bbb S$ is said to be **pairwise disjoint** iff:

- $\forall X, Y \in \Bbb S: X \ne Y \implies X \cap Y = \varnothing$

Here, $\cap$ denotes intersection, and $\varnothing$ denotes the empty set.

Alternatively, we can say that the elements of $\Bbb S$ are **pairwise disjoint**.

### Family of Sets

An indexed family of sets $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ is said to be **pairwise disjoint** if and only if:

- $\forall i, j \in I: i \ne j \implies S_i \cap S_j = \varnothing$

Hence the indexed sets $S_i$ themselves, where $i \in I$, are referred to as being **pairwise disjoint**.

## Also known as

Other names for **pairwise disjoint** include **mutually disjoint** and **non-intersecting**.