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Cartesian product of the sets {x,y,z} and {1,2,3}In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1]
In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
In the mathematical field of graph theory, the intersection number of a graph = (,) is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element.
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). graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
f −1 [B 0] ↪ A. and the restriction of f to f −1 [B 0] f −1 [B 0] → B 0. Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under f of the subobject specified by g. Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the ...
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The line graph of a graph G is defined as the intersection graph of the edges of G, where we represent each edge as the set of its two endpoints. A string graph is the intersection graph of curves on a plane. A graph has boxicity k if it is the intersection graph of multidimensional boxes of dimension k, but not of any smaller dimension.