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In mathematics, the disjoint union (or discriminated union) of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.
The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and f i : X i → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute:
In graph theory, a cograph, or complement-reducible graph, or P 4-free graph, is a graph that can be generated from the single-vertex graph K 1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K 1 and is closed under complementation and disjoint union.
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
The coproduct in the category of sets is simply the disjoint union with the maps i j being the inclusion maps.Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be ...
λ-system (Dynkin system) – Family closed under complements and countable disjoint unions; π-system – Family of sets closed under intersection; Ring of sets – Family closed under unions and relative complements; Russell's paradox – Paradox in set theory (or Set of sets that do not contain themselves)
The important special case where the family of maps consists of a single surjective map can be completely characterized using the notion of quotient map.A surjective function : (,) (,) between topological spaces is a quotient map if and only if the topology on coincides with the final topology induced by the family = {}.
In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some <) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the ( n − 1 ) {\displaystyle (n-1)} -dimensional sphere ) to elements of the k {\displaystyle k} -dimensional ...