Difference between revisions of "Cartesian product"
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Latest revision as of 14:25, 19 February 2016
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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