# Cartesian product

From Maths

## Contents

## Definition

Given two sets, [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] their *Cartesian product* is the set:

- [ilmath]X\times Y:=\{(x,y)\ \vert\ x\in X\wedge y\in Y\}[/ilmath], note that [ilmath](x,y)[/ilmath] is an
*ordered pair*traditionally this means- [ilmath](x,y):=\{\{x\},\{x,y\}\}[/ilmath] or indeed
- [ilmath]X\times Y:=\Big\{\{\{x\},\{x,y\}\}\ \vert\ x\in X\wedge y\in Y\Big\}[/ilmath]

### Set construction

TODO: Build a set that contains [ilmath]\{x,y\} [/ilmath]s, then build another that contains ordered pairs, then the Cartesian product is a subset of this set

### Projections

With the *Cartesian product* of [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] come two maps:

- [ilmath]\pi_1:X\times Y\rightarrow X[/ilmath] given by [ilmath]\pi_1:(x,y)\mapsto x[/ilmath] and
- [ilmath]\pi_2:X\times Y\rightarrow Y[/ilmath] given by [ilmath]\pi_2:(x,y)\mapsto y[/ilmath]

TODO: Give explicitly

## Properties

The *Cartesian product* has none of the usual^{[Note 1]} properties:

Property | Definition | Meaning | Comment |
---|---|---|---|

Associativity | [ilmath]X\times(Y\times Z)=(X\times Y)\times Z[/ilmath] | No | We can side-step this with obvious mappings |

Commutativity | [ilmath]X\times Y=Y\times X[/ilmath] | No |

### Associativity

Given [ilmath]X[/ilmath], [ilmath]Y[/ilmath] and [ilmath]Z[/ilmath] notice the two ways of interpreting the *Cartesian product* are:

- [ilmath](X\times Y)\times Z[/ilmath] which gives elements of the form [ilmath]((x,y),z)[/ilmath] and
- [ilmath]X\times (Y\times Z)[/ilmath] which gives elements of the form [ilmath](x,(y,z))[/ilmath]

It is easy to construct a bijection between these, thus it rarely matters.

## Notes

- ↑ By usual I mean common properties of binary operators, eg associativity, commutative sometimes, so forth

## References

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