Search results
Results from the WOW.Com Content Network
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be ...
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) —where a ∈ A and b ∈ B. [5] The class of all things (of a given type) that have Cartesian products is called a Cartesian ...
A universe set is an absorbing element of binary union . The empty set is an absorbing element of binary intersection and binary Cartesian product , and it is also a left absorbing element of set subtraction :
Furthermore, given a set , the product order over the Cartesian product {,} can be identified with the inclusion ordering of subsets of . [4] The notion applies equally well to preorders . The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras .
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology , which can also be given to a product space and which agrees ...
Then, for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either A has Lebesgue measure 0 or B is a single point. (In this case the ...
The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B. A binary relation between sets A and B is a subset of A × B. The (a, b) notation may be used for other purposes, most notably as denoting open intervals on the real number line ...
A Cartesian product of two graphs. In graph theory, the Cartesian product G H of graphs G and H is a graph such that: the vertex set of G H is the Cartesian product V(G) × V(H); and; two vertices (u,v) and (u' ,v' ) are adjacent in G H if and only if either u = u' and v is adjacent to v' in H, or; v = v' and u is adjacent to u' in G.