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Encyclopedia > Cartesian product

In mathematics, the Cartesian product is a direct product of sets. Specifically, the Cartesian product of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...

Xtimes Y = {(x,y) | xin X;mathrm{and};yin Y}.

The Cartesian product is named after René Descartes whose formulation of analytic geometry gave rise to this concept. René Descartes (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, and scientist. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ...


Concretely, if set X is the 13-element set of ranks { A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2 } and set Y is the 4-element set of suits {♠, ♥, ♦, ♣}, then the Cartesian product of those two sets is the 52-element set of standard playing cards { (A, ♠), (K, ♠), ..., (2, ♠), (A, ♥), ..., (3, ♣), (2, ♣) }.

Contents

Cartesian square and n-ary product

The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X2 = X × X. An example is the 2-dimensional plane R2 = R × R where R is the set of real numbers - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system). Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ... In mathematics, the real numbers may be described informally in several different ways. ... Fig. ...


This can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn:

Indeed, it can be identified to (X1 × ... × Xn-1) × Xn. It is a set of n-tuples. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...


An example of this is the Euclidean 3-space R3 = R × R × R, with R again the set of real numbers. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...


As an aid to its calculation, a table can be drawn up, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table by choosing the element of the set from the row and the column.


Infinite products

The above definition is usually all that's needed for the most common mathematical applications. However, it is possible to define the Cartesian product over an arbitrary (possibly infinite) collection of sets. If I is any index set, and Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, an index set is another name for a function domain. ...

{X_i | i in I}

is a collection of sets indexed by I, then we define

prod_{i in I} X_i = { f : I to bigcup_{i in I} X_i | (forall i)(f(i) in X_i)},

that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of Xi . In mathematics, an index set is another name for a function domain. ...


For each j in I, the function

defined by

is called the j th projection map.


An n-tuple can be viewed as a function on {1, 2, ..., n} that takes its value at i to be the i th element of the tuple. Hence, when I is {1, 2, ..., n} this definition coincides with the definition for the finite case. In the infinite case this is a family. In mathematics, an index set is another name for a function domain. ...


One particular and familiar infinite case is when the index set is mathbb N, the natural numbers: this is just the set of all infinite sequences with the i th term in its corresponding set X. Once again, mathbb R provides an example of this: Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

is the collection of infinite sequences of real numbers, and it is easily visualized as a vector or tuple with an infinite number of components. Another special case (the above example also satisfies this) is when all the factors Xi involved in the product are the same, being like "Cartesian exponentiation." Then the big union in the definition is just the set itself, and the other condition is trivially satisfied, so this is just the set of all functions from I to X.


Otherwise, the infinite cartesian product is less intuitive; though valuable in its applications to higher mathematics.


The assertion that the Cartesian product of arbitrary non-empty collection of non-empty sets is non-empty is equivalent to the axiom of choice. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...


Abbreviated form

If several sets are being multiplied together, e.g. X1,X2,X3,..., then some authors [1] choose to abbreviate the Cartesian product as simply .


Cartesian product of functions

If f is a function from A to B and g is a function from X to Y, their cartesian product f×g is a function from A×X to B×Y with

(ftimes g)(a, x) = (f(a), g(x))

As above this can be extended to tuples and infinite collections of functions. In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...


Category theory

Categorically, the cartesian product is the product in the Category of sets. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...


See also

In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ... In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ...

External links

  • Cartesian Product at ProvenMath

References

  1. ^ M. Osborne and A. Rubinstein, A Course in Game Theory, MIT Press 1994.

  Results from FactBites:
 
No More Headaches with the Cartesian Product @ POWERBUILDER DEVELOPER'S JOURNAL (2067 words)
Cartesian product n (1958) : a set that is constructed from two given sets and comprises all pairs of elements such that the first element of the pair is from the first set and the second is from the second set (Merriam Webster's Collegiate Dictionary, Tenth Edition)
The number of rows in a Cartesian product of two tables is equal to the number of rows in the first table times the number of rows in the second table.
The Cartesian product of the employee and department tables is 375 (75 * 5).
SQL Cartesian Joins - FluffyCat.com (413 words)
A Cartesian join is when you join every row of one table to every row of another table.
Note: To make your own Cartesian join at home all you really need is to take your basic select with two or more tables and remove the where clause.
In general a cartesian join is very resource intensive, and it's usually done by accident.
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