Pairwise disjoint

English

$A$, $B$, and $C$ are sets with elements represented by distinctly colored circles. The sets are pairwise disjoint since the intersection of any two distinct sets is empty.

Adjective

1. Template:Lb Let $\{A_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ be any collection of sets indexed by a set $\mathrm{\Lambda }$. We call the indexed collection pairwise disjoint if for any two distinct indices, $\lambda ,\mu \in \mathrm{\Lambda }$, the sets $A_{\lambda }$ and $A_{\mu }$ are disjoint.
   • Template:Quote-book
   • 2009, John M. Franks, A (Terse) Introduction to Lebesgue Integration, Template:W, page 27,

     For example, if we had a collection of pairwise disjoint intervals of length $1/2,1/4,1/8,\dots 1/2^n,\dots$,etc., then we would certainly like to be able to say that the measure of their union we is the sum $\sum 1/2^n=1$ which would not follow from finite additivity.

   • Template:Quote-book

Usage notes

The condition is a generalization of the concept of disjoint sets, from two to an arbitrary collection of sets. When applied to a collection, the original formulation - that the sets have an intersection equal to the empty set - becomes ambiguous and in need of clarification.

Synonyms

• Template:Sense mutually disjoint

Translations

• Catalan: Template:T, Template:T
• Dutch: Template:T, Template:T
• French: Template:T, Template:T
• German: Template:T
• Indonesian: Template:T
• Italian: Template:T, Template:T
• Mandarin: Template:T
• Spanish: Template:T, Template:T

Further reading

• Template:Pedia
• Template:Pedia