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): 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