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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 branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets , and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the ...
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of the form X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X. For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ P the closed sets of X.
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 this way, the disjoint union construction provides a way of viewing any family of sets indexed by as a set "fibered" over , and conversely, for any set : fibered over , we can view it as the disjoint union of the fibers of . Jacobs has referred to these two perspectives as "display indexing" and "pointwise indexing".
In pseudocode, union by rank is: function Union(x, y) is // Replace nodes by roots x := Find(x) y := Find(y) if x = y then return // x and y are already in the same set end if // If necessary, rename variables to ensure that // x has rank at least as large as that of y if x.rank < y.rank then (x, y) := (y, x) end if // Make x the new root y ...
where the denotes the disjoint union, and ~ is the equivalence relation generated by ( 0 , x ) ∼ f ( x ) for each x ∈ X . {\displaystyle (0,x)\sim f(x)\quad {\text{for each }}x\in X.} That is, the mapping cylinder M f {\displaystyle M_{f}} is obtained by gluing one end of X × [ 0 , 1 ] {\displaystyle X\times [0,1]} to Y {\displaystyle Y ...