Cartesian product

Definition

Given two sets, $X$ and $Y$ their Cartesian product is the set:

• $X\times Y:=\{(x,y)\ \vert\ x\in X\wedge y\in Y\}$, note that $(x,y)$ is an ordered pair traditionally this means
  • $(x,y):=\{\{x\},\{x,y\}\}$ or indeed
  • $X\times Y:=\Big\{\{\{x\},\{x,y\}\}\ \vert\ x\in X\wedge y\in Y\Big\}$

Set construction

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TODO: Build a set that contains $\{x,y\}$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 $X$ and $Y$ come two maps:

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

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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	$X\times(Y\times Z)=(X\times Y)\times Z$	No	We can side-step this with obvious mappings
Commutativity	$X\times Y=Y\times X$	No

Associativity

Given $X$, $Y$ and $Z$ notice the two ways of interpreting the Cartesian product are:

• $(X\times Y)\times Z$ which gives elements of the form $((x,y),z)$ and
• $X\times (Y\times Z)$ which gives elements of the form $(x,(y,z))$

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

References

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