Cartesian product
In
mathematics, the
Cartesian product (or product set) is a
direct product of sets. The Cartesian product is named after
René Descartes, whose formulation of
analytic geometry gave rise to this concept. Specifically, the Cartesian product of two
sets X (for example the points on an x-axis) and
Y (for example the points on a y-axis), 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 (e.g. the whole of the x-y plane):
X⋅ Y = (xy) || ξn X;mathrmand;y∈ Y.
For example, the Cartesian product of the thirteen-element set of standard playing card ranks {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2} and the four-element set of card suits {♠, ♥, ♦, ♣} is the 52-element set of playing cards {(Ace, ♠), (King, ♠), ..., (2, ♠), (Ace, ♥), ..., (3, ♣), (2, ♣)}. The Cartesian product has 52 elements because that is the product of 13 times 4.A Cartesian product of two finite sets can be represented by a table, 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.
n-ary product
The Cartesian product can be generalized to the
n-ary Cartesian product over
n sets
X1, ...,
Xn:
X1⋅cderiv(⋅)s⋅ Xn = (x1 lderiv(⋅)s xn) mid x1∈ X1;mathrmand;cderiv(⋅)s;mathrmand;xn∈ Xn.
Indeed, it can be identified to (
X1 × ... ×
Xn-1) ×
Xn. It is a set of
n-tuples.
Cartesian square and Cartesian power
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).The
cartesian power of a
set X can be defined as:
Xn = X ⋅ X cderiv(⋅)s⋅ X n= ( (x1 x2 lderiv(⋅)s xn) || x1 ∈ X ∧ x2 ∈ X ∧ lderiv(⋅)s xn ∈ X &nbs(;).
An example of this is
R3 =
R ×
R ×
R, with
R again the set of real numbers, and more generally
Rn.See also:
- Euclidean space
- {{ml|total order|Orders on the Cartesian product of totally ordered sets|orders on Rn}}
Infinite products
It is possible to define the Cartesian product of an arbitrary (possibly
infinite)
family of sets. If
I is any
index set, and {
Xi |
i ∈
I} is a collection of sets indexed by
I, then the Cartesian product of the sets in
X is defined to be
&(rod;i ∈ I Xi = f : I → big&cu(;i ∈ I Xi || (∀ i)(f(i) ∈ Xi)
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 .For each
j in
I, the function
&(i;j : &(rod;i ∈ I Xi → Xj
defined by
πj(
f) =
f(
j) is called the
j -th projection map.An important case is when the index set is
N the
natural numbers: this Cartesian product is the set of all infinite sequences with the
i -th term in its corresponding set
Xi . For example, each element of
&(rod;n = 1&∈f∈; R =Rω= R ⋅ R ⋅ cderiv(⋅)s
can be visualized as a vector with an infinite number of real-number components. The special case
Cartesian exponentiation occurs when all the factors
Xi involved in the product are the same set
X. In this case,
&(rod;i ∈ I Xi = &(rod;i ∈ I X
is the set of all functions from
I to
X. This case is important in the study of
cardinal exponentiation. The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, 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 (in some settings, this is taken as the very definition of an
n-tuple).Nothing in the definition of infinite Cartesian products implies that the Cartesian product of nonempty sets must itself be nonempty. This assertion is equivalent to the
axiom of choice.
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
×Xi.
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
(f⋅ g)(a x) = (f(a) g(x))
As above this can be extended to
tuples and infinite collections of functions.
Category theory
Although the Cartesian product is traditionally applied to sets,
category theory provides a more general interpretation of the
product of mathematical structures. This is distinct from, although related to, the notion of a
Cartesian square in category theory, which is a generalization of a
fiber product.
Graph theory
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 is adjacent to
u′ and
v is adjacent to
v′. Unlike the ordinary Cartesian product, the Cartesian product of graphs is not a
product in the sense of category theory. Instead it is more like a
tensor product.
See also
External links
References
-
[M. Osborne and A. Rubinstein, A Course in Game Theory, MIT Press 1994.]
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