Difference between revisions of "Function"
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==Notation== | ==Notation== | ||
| − | A function {{M|f}} from a domain {{M|X}} to a set {{M|Y}} is denoted {{M|f:X\rightarrow Y}} | + | * A function {{M|f}} from a domain {{M|X}} to a set {{M|Y}} is denoted {{M|f:X\rightarrow Y}} |
| + | * If we have say two [[Topological space|topological spaces]] {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} then we may write: | ||
| + | ** {{M|f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})}} and mean {{M|f:X\rightarrow Y}} | ||
| + | * '''That is to say that as a general rule given a function {{M|f:(A_1,A_2,\cdots)\rightarrow(B_1,B_2,\cdots)}} take it as a function {{M|f:A_1\rightarrow B_1}}''' | ||
| + | ** A [[Tuple|tuple]] makes no sense there anyway, for multiple arguments we use the [[Cartesian product]] anyway. | ||
{{Todo|Come back after the relation page and fill this out}} | {{Todo|Come back after the relation page and fill this out}} | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
Revision as of 14:26, 16 June 2015
A function [ilmath]f[/ilmath] is a special kind of relation
Contents
Domain
A function ought be defined for everything in its domain, that's for every point in the domain the function maps the point to something.
Examples
(See notation below if you're not sure what the [math]f:X\rightarrow Y[/math] notation means)
- [math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=\frac{1}{x}[/math] isn't defined at [math]0[/math]
- [math]f:\mathbb{R}\rightarrow\mathbb{R}[/math] given by [math]f(x)=x^2[/math] is correct, it is not surjective though, because nothing maps onto the negative numbers, however [math]f:\mathbb{R}\rightarrow\mathbb{R}_{\ge 0}[/math] with [math]f(x)=x^2[/math] is a surjection. It is not an injective function as only [math]0[/math] maps to one point.
Notation
- A function [ilmath]f[/ilmath] from a domain [ilmath]X[/ilmath] to a set [ilmath]Y[/ilmath] is denoted [ilmath]f:X\rightarrow Y[/ilmath]
- If we have say two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] then we may write:
- [ilmath]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/ilmath] and mean [ilmath]f:X\rightarrow Y[/ilmath]
- That is to say that as a general rule given a function [ilmath]f:(A_1,A_2,\cdots)\rightarrow(B_1,B_2,\cdots)[/ilmath] take it as a function [ilmath]f:A_1\rightarrow B_1[/ilmath]
- A tuple makes no sense there anyway, for multiple arguments we use the Cartesian product anyway.
TODO: Come back after the relation page and fill this out
