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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:
The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated ...
The 2-regular graphs are the disjoint unions of cycle graphs. [4] More generally, every graph is the disjoint union of connected graphs, its connected components. The cographs are the graphs that can be constructed from single-vertex graphs by a combination of disjoint union and complement operations. [5]
Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
Geometric join of two line segments.The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, in gray.. In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in .
graph union: G 1 ∪ G 2. There are two definitions. In the most common one, the disjoint union of graphs, the union is assumed to be disjoint. Less commonly (though more consistent with the general definition of union in mathematics) the union of two graphs is defined as the graph (V 1 ∪ V 2, E 1 ∪ E 2).
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.