# Definition:Pairwise Disjoint

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## Definition

### Set of Sets

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

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

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

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

### Family of Sets

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

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

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

## Also known as

A set of sets whose elements are **pairwise disjoint** is often referred to as a **pairwise disjoint collection**.

Other names for **pairwise disjoint** include:

Some sources hyphenate: **pair-wise disjoint**.

When the collection in question is implemented as an indexed family the compact term **disjoint family** is often seen.

## Also see

- Results about
**pairwise disjoint**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**pair-wise disjoint** - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**pair-wise disjoint** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**mutually disjoint**