Definition:Cartesian Product

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[edit] Definition

The cartesian product (or Cartesian product) of two sets S and T is the set of ordered pairs \left({x, y}\right) with x \in S and y \in T.

This is denoted:

S \times T = \left\{{\left({x, y}\right) : x \in S \and y \in T}\right\}

Some authors call this the direct product of S and T.

Some call it the cartesian product set, others just the product set.


In a cartesian product S \times T, the sets S and T are called the factors of S \times T.


Another way of defining it is by:

\left({x, y}\right) \in S \times T \iff x \in S, y \in T


It is also known as the cross product of two sets, but this can be confused with other usages of this term.


[edit] Generalized Definition

Let \left \langle {S_n} \right \rangle be a sequence of sets.

The cartesian product of \left \langle {S_n} \right \rangle is defined as:

\prod_{k=1}^n S_k = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S_k}\right\}


It is also denoted S_1 \times S_2 \times \ldots \times S_n.

Thus S_1 \times S_2 \times \ldots \times S_n is the set of all ordered n-tuples \left({x_1, x_2, \ldots, x_n}\right) with x_k \in S_k.


[edit] Cartesian Space

Let S be a set.

Then the cartesian nth power of S, or S to the power of n, is defined as:

S^n = \prod_{k=1}^n S = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S}\right\}


Thus S^n = S \times S \times \ldots \left({n}\right) \ldots \times S

Alternatively it can be defined recursively:

S^n = \begin{cases} S: & n = 1 \\ S \times S^{n-1} & n > 1 \end{cases}


The set S^n called a cartesian space.


An element x_j of a tuple \left({x_1, x_2, \ldots, x_n}\right) of a cartesian space S^n is known as a basis element of S^n.


[edit] Real Cartesian Space

When S is the set of real numbers \R, the cartesian product takes on a special significance.


Let n \in \N^*.

Then \R^n is the cartesian product defined as follows:

\R^n = \R \times \R \times \cdots \left({n}\right) \cdots \times \R = \prod_{k=1}^n \R


Similarly, \R^n can be defined as the set of all real n-tuples:

\R^n = \left\{{\left({x_1, x_2, \ldots, x_n}\right): x_1, x_2, \ldots, x_n \in \reals}\right\}

It can be shown that:


[edit] Notes

The notation for the cartesian power of a set S^n should not be confused with the notation used for the conjugate of a set.

Also beware not to confuse the name of the concept itself with that of the power set \mathcal{P} \left({S}\right) of S.


[edit] Source of Name

This entry was named for René Descartes.


[edit] Sources

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