pairwise disjoint

English

{\\displaystyle A}

,

{\\displaystyle B}

, and

{\\displaystyle 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

pairwise disjoint (not comparable)

(mathematics, set theory, of a collection of two or more sets) Let ${\\displaystyle \\{A_{\\lambda }\\}_{\\lambda \\in \\Lambda }}$ $\text{[math]}$ be any collection of sets indexed by a set ${\\displaystyle \\Lambda }$ $\text{[math]}$ . We call the indexed collection pairwise disjoint if for any two distinct indices, ${\\displaystyle \\lambda ,\\mu \\in \\Lambda }$ $\text{[math]}$ , the sets ${\\displaystyle A_{\\lambda }}$ $\text{[math]}$ and ${\\displaystyle A_{\\mu }}$ $\text{[math]}$ are disjoint.
- 2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2nd Edition, 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

Translations

such that any two distinct sets are disjoint