# Cartesian Product Constructor and Functions

The special constructor `car< ... >` is used for the creation of cartesian products of structures.

## `car< R₁, ..., Rₖ >: Str, ..., Str -> SetCart`

Given a list of sets or algebraic structures $R_1, \ldots, R_k$, construct the cartesian product set $R_1 \times \cdots \times R_k$.

## `CartesianProduct(R, S): Str, ..., Str -> SetCart`

Given structures $R$ and $S$, construct the cartesian product set $R \times S$. This is the same as calling the `car` constructor with the two arguments $R$ and $S$.

## `CartesianProduct(L): [Str] -> SetCart`

## `CartesianProduct(L): <Str> -> SetCart`

Given a sequence or tuple $L$ of structures, construct the cartesian product of the elements of $L$.

## `CartesianPower(R, k): Str, RngIntElt -> SetCart`

Given a structure $R$ and an integer $k$, construct the cartesian power set $R^{k}$.

## `Flat(C): SetCart -> SetCart`

Given a cartesian product $C$ of structures which may themselves be cartesian products, return the cartesian product of the base structures, considered in depth-first order (see [`Flat`](tuple.md#function-tuple-flat2) for the element version).

## `NumberOfComponents(C): SetCart -> RngIntElt`

Given a cartesian product $C$, return the number of components of $C$.

## `Component(C, i): SetCart, RngIntElt -> Str`

## `C[i]: SetCart, RngIntElt -> Str`

The $i$-th component of $C$.

## `Components(C): SetCart -> List`

The list of components of a cartesian product.

## `# C: SetCart -> RngIntElt`

Given a cartesian product $C$, return the cardinality of $C$.

## `Rep(C): SetCart -> Elt`

Given a cartesian product $C$, return a representative of $C$.

## `Random(C): SetCart -> Elt`

Given a cartesian product $C$, return a random element of $C$.

## `Example: Cartesian Product (ex-c2f16a)`

We create the product of ${\mathbb{Q}}$ and ${\mathbb{Z}}$.

```magma
> C := car< RationalField(), Integers() >;
> C;
Cartesian Product<Rational Field, Ring of Integers>

```