# 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

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

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

## Also see

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**pair-wise disjoint** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**pair-wise disjoint** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**mutually disjoint**