Disjoint

A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.

But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets. citation.

A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them.

Interior-disjoint shapes The red shapes are not interior-disjoint the blue Triangle.

It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself. citation.

The green and the yellow shapes are interior-disjoint with the blue Triangle, but only the yellow shape is entirely disjoint from the blue Triangle.

Two cycles are disjoint if they move disjoint subsets of elements.

A disjoint union may mean one of two things.

A family F of sets is pairwise disjoint if, for every two sets in the family, their intersection is empty.

Also the empty family of sets is pairwise disjoint.

A topological space X is disconnected if there exist disjoint, nonempty, closed subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.

Axiom of infinity Ramified types and the axiom of reducibility In simple type theory objects are elements of various disjoint "types".

Definition Countable additivity of a measure μ : The measure of a countable disjoint union is the same as the sum of all measures of each subset.

Disadvantages Stratified sampling is not useful when the population cannot be exhaustively partitioned into disjoint subgroups.

Disjoint-set data structures citation.

For example, the disjoint union of a Sierpinski carpet and a circle is also a universal plane curve.

For instance, the closed intervals of the real numbers form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint. citation.

For instance, two infinite sets whose intersection is a finite set may be said to be almost disjoint. citation.

For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set. citation.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y).