# pairwise disjoint

## English

pairwise disjoint (not comparable)

1. (mathematics, set theory, of a collection of two or more sets) Let ${\displaystyle \{A_{\lambda }\}_{\lambda \in \Lambda }}$ be any collection of sets indexed by a set ${\displaystyle \Lambda }$. We call the indexed collection pairwise disjoint if for any two distinct indices, ${\displaystyle \lambda ,\mu \in \Lambda }$, the sets ${\displaystyle A_{\lambda }}$ and ${\displaystyle A_{\mu }}$ are disjoint.
• 2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 61:
Proposition 4.5. Every permutation is a product of pairwise disjoint cycles, and this decomposition is unique up to the order of the terms.
• 2009, John M. Franks, A (Terse) Introduction to Lebesgue Integration, American Mathematical Society, page 27,
For example, if we had a collection of pairwise disjoint intervals of length ${\displaystyle 1/2,1/4,1/8,\dots 1/2^{n},\dots }$,etc., then we would certainly like to be able to say that the measure of their union we is the sum ${\displaystyle \sum 1/2^{n}=1}$ which would not follow from finite additivity.
• 2015, Su Gao, Stephen C Jackson, Brandon Seward, Group Colorings and Bernoulli Subflows, American Mathematical Society, page 158:
To show that all ${\displaystyle \Gamma _{i}}$-translates of ${\displaystyle F_{i}}$, are pairwise disjoint, it suffices to show that all ${\displaystyle \Gamma _{i,0}}$-translates of ${\displaystyle F_{i}}$ are pairwise disjoint, since then the argument as above will show inductively that the ${\displaystyle \Gamma _{i,m}}$-translates of ${\displaystyle F_{i}}$ are pairwise disjoint for all ${\displaystyle m>0}$.

#### 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

• (such that any two distinct sets are disjoint): mutually disjoint