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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 ...
The Cartesian square of a set X is the Cartesian product X 2 = X × X. An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers : [ 1 ] R 2 is the set of all points ( x , y ) where x and y are real numbers (see the Cartesian coordinate system ).
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.
The product category C × D has: as objects: pairs of objects (A, B), where A is an object of C and B of D; as arrows from (A 1, B 1) to (A 2, B 2): pairs of arrows (f, g), where f : A 1 → A 2 is an arrow of C and g : B 1 → B 2 is an arrow of D; as composition, component-wise composition from the contributing categories: (f 2, g 2) o (f 1 ...
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]
This is known as triple product expansion, or Lagrange's formula, [2] [3] although the latter name is also used for several other formulas. Its right hand side can be remembered by using the mnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided below.
The product X×Y is the Cartesian product of X and Y, and Z Y is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X × Y → Z is naturally identified with the curried function g : X → Z Y defined by g ( x )( y ) = f ( x , y ) for all x in X and y in Y .
For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}. It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A). Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a, b), where a is a member of A and b is a ...