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One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
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 ...
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 .
The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: given sets A i where i ranges over some index set I, we construct the coproduct as the union of A i ×{i} (the cartesian product with i serves to ensure that all the components stay disjoint).
The lexicographical order defines an order on an n-ary Cartesian product of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a Cartesian product E 1 × ⋯ × E n {\displaystyle E_{1}\times \cdots \times E_{n}} is a sequence whose i {\displaystyle i} th element belongs to E i {\displaystyle ...
In mathematics, one can often define a direct product of objects already known, giving a new one. This induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. More abstractly, one talks about the product in category theory, which formalizes these notions.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
[1] (Of course, Kőnig's theorem is trivial if the cardinal numbers m i and n i are finite and the index set I is finite. If I is empty, then the left sum is the empty sum and therefore 0, while the right product is the empty product and therefore 1.) Kőnig's theorem is remarkable because of the strict inequality in the conclusion.