Cartesian product

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TODO: Find references

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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

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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

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Projections

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

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

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TODO: Give explicitly

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Properties

The Cartesian product has none of the usual 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.