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The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order. [3] The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. [7]
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 represented by an n-dimensional array, where each element is an n-tuple.
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.
A product of two or more non-trivial rings always has nonzero zero divisors: if x is an element of the product whose coordinates are all zero except p i (x) and y is an element of the product with all coordinates zero except p j (y) where i ≠ j, then xy = 0 in the product ring.
Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
If A and B are sets, then the Cartesian product (or simply product) is defined to be: A × B = {(a,b) | a ∈ A and b ∈ B}. That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B.
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 ...
In the above examples, the cardinality of the set A is 4, while the cardinality of set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets.
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