Search results
Results From The WOW.Com Content Network
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 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]
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.
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".
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).
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 ...
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 ...