View source for Function (notation)

'''''This page describes the notation of how we use [[function|functions]] for information on what a function is, see [[function]]'''''

==Basics==
* {{M|f:X\rightarrow Y}} - the most basic form, here {{M|f}} is a [[relation]] that associates with each {{M|x\in X}} a {{M|y\in y}}. We write this as {{M|1=y=f(x)}}

==Abuses of notation==
* '''Tuples: ''' sometimes we will write {{M|f:(X,A)\rightarrow(Y,B)}}, this simply means that {{M|f:X\rightarrow Y}} where {{M|X}} is some sort of space (with structure {{M|A}}) and {{M|Y}} is some sort of space with a structure {{M|B}}.
** Possible misinterpretation: 
**: {{M|f:X\times A\rightarrow Y\times B}} denotes a function, {{M|f}} that takes ordered pairs, {{M|(x,a)\in X\times A}} to ordered pairs, {{M|(y,b)\in Y\times B}}, this notation clearly operates on sets (as it uses the [[Cartesian product]]) keeping with the convention of the thing either side of the {{M|\rightarrow}} is a set. So the notation {{M|f:(X,A)\rightarrow(Y,B)}} for {{M|f}}'s input being a tuple, {{M|(x,a)}} is ''absurd'' because:
**:# It is another notation for something we already have
**:# It violates the "sets being either side of the arrow" thing ({{M|A\times B}} is a set, {{M|(\cdot,\cdot)}}, even if considered as an [[Ordered pair]] does not "evaluate" to something useful when it comes to relations. 
** '''Warnings:'''
**# Sometimes {{M|f:(X,A)\rightarrow(Y,B)}} denotes<ref name="FOAT">Fundamentals of Algebraic Topology - Steven H. Weintraub</ref> that {{M|f:X\rightarrow Y}} with the additional information of {{M|f\vert_A:A\rightarrow B}}, or more simply is to say that {{M|f:X\rightarrow Y}} with the additional statement: {{M|f(A)\subseteq B}}
**#* Use this if {{M|(X,A)}} has not previously been declared as some sort of space. 
**#* {{Note|I've only ever seen this used in one book - Fundamentals of Algebraic Topology by Steven H. Weintraub}}
** Examples:
**# Let {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} be [[Topological space|topological spaces]], let {{M|f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})}} be a [[Continuous map|continuous map]]...
**#* Here the tuples (as usual) help the reader/writer keep track of spaces, in this case the topologies on {{M|X}} and {{M|Y}}
**#* This example extends to [[Measurable space|measurable spaces]], [[Vector space|vector spaces]] and many more.

==References==
<references/>