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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:
For symmetric difference, the sets ( ) and () = ( ) are always disjoint. So these two sets are equal if and only if they are both equal to ∅ . {\displaystyle \varnothing .} Moreover, L ∖ ( M R ) = ∅ {\displaystyle L\,\setminus \,(M\,\triangle \,R)=\varnothing } if and only if L ∩ M ∩ R = ∅ and L ⊆ M ∪ R . {\displaystyle L\cap M ...
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.
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 ...
1. The asymmetric relation between two adjacent vertices in a graph, represented as an arrow. 2. The asymmetric relation between two vertices in a directed path. disconnect Cause to be disconnected. disconnected Not connected. disjoint 1. Two subgraphs are edge disjoint if they share no edges, and vertex disjoint if they share no vertices. 2.
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
We say that and are disjoint if does not intersect . In plain language, they have no elements in common. A {\displaystyle A} and B {\displaystyle B} are disjoint if their intersection is empty , denoted A ∩ B = ∅ . {\displaystyle A\cap B=\varnothing .}