Definition:Pairwise Disjoint/Set of Sets

Definition

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.

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.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.20$: Decomposition of a Set: Definition $20.1$