Cartesian product

productCartesian squareCartesian power×product setproductsCartesian product algorithmcross productcrossedcylinder
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.wikipedia
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Direct product

internal direct product(direct) productsdirectly indecomposable
The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set.

Set theory

axiomatic set theoryset-theoreticset
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.
.*Cartesian product of

Ordered pair

ordered pairspairpairs
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.
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs.

Set-builder notation

set builder notationabstractionbuild the sets
In terms of set-builder notation, that is
Here the cartesian product denotes the set of ordered pairs of real numbers.

Indexed family

familyindicesindex
More generally still, one can define the Cartesian product of an indexed family of sets.
Likewise for intersections and cartesian products.

Tuple

tuplesn-tuple5-tuple
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.
-fold Cartesian power

Axiom of choice

Choicechoice principleschosen
Even if each of the X i is nonempty, the Cartesian product may be empty if the axiom of choice (which is equivalent to the statement that every such product is nonempty) is not assumed.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty.

Plane (geometry)

planeplanarplanes
An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers: R 2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
This section is solely concerned with planes embedded in three dimensions: specifically, in R 3.

René Descartes

DescartesCartesianRene Descartes
The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates.

Axiom of power set

power setpower set axiomAxiom of the power set
Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification.
The Power Set Axiom allows a simple definition of the Cartesian product of two sets X and Y:

Product (category theory)

productproductscategorical product
Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures.
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.

Complement (set theory)

complementset differencecomplements
: where denotes the absolute complement of A.
A binary relation R is defined as a subset of a product of sets X × Y.

Codomain

valuesimagetarget
As a special case, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
, with F a functional subset of the Cartesian product

Real number

realrealsreal-valued
An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers: R 2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system). In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates.
The notation R n refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter.

Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers: R 2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system). The main historical example is the Cartesian plane in analytic geometry.
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is with the Cartesian product, where \R is the set of all real numbers.

Binary relation

relationrelationsidentity relation
Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
That is, it is a subset of the Cartesian product A × B. It encodes the information of relation: an element a is related to an element b if and only if the pair (a, b) belongs to the set.

Cartesian closed category

Cartesian closed categoriescartesian closedcartesian-closed category
Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category.
Named after (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.

Infinite set

infiniteinfinitelyinfinitely many
The set A × B is infinite if either A or B is infinite and the other set is not the empty set.
The Cartesian product of an infinite set and a nonempty set is infinite.

Finitary relation

relationrelationsTheory of relations
The relation itself is a mathematical object defined in terms of concepts from set theory (i.e., the relation is a subset of the Cartesian product on {Person X, Person Y, Person Z}), that carries all of the information from the table in one neat package.

Function (mathematics)

functionfunctionsmathematical function
As a special case, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X. Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
The Cartesian product of n sets is the set of all n-tuples such that x_i\in X_i for every i with.

Cartesian product of graphs

Cartesian productCartesian productscartesian products of graphs
In graph theory the Cartesian product of two graphs G and H is the graph denoted by G × H whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G.

Product topology

productproduct spacetopological product
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.

Join (SQL)

joinjoinsInner join
CROSS JOIN returns the Cartesian product of rows from tables in the join.

Subset

supersetproper subsetsubsets
Other properties related with subsets are:
For the power set of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product of k = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1.