Definition:Pairwise Disjoint

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:

mutually disjoint

non-intersecting.

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

• Definition:Disjoint Sets

• 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