Difference between revisions of "Relation"
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Revision as of 18:40, 1 March 2015
A set [ilmath]R[/ilmath] is a binary relation if all elements of [ilmath]R[/ilmath] are ordered pairs. That is for any [ilmath]z\in R\ \exists x\text{ and }y:(x,y)[/ilmath]
Contents
Notation
Rather than writing [ilmath](x,y)\in R[/ilmath] to say [ilmath]x[/ilmath] and [ilmath]y[/ilmath] are related we can instead say [ilmath]xRy[/ilmath]
Domain
The set of all [ilmath]x[/ilmath] which are related by [ilmath]R[/ilmath] to some [ilmath]y[/ilmath] is the domain.
[math]\text{Dom}(R)=\{x|\exists\ y: xRy\}[/math]
Range
The set of all [ilmath]y[/ilmath] which are a relation of some [ilmath]x[/ilmath] by [ilmath]R[/ilmath] is the range.
[math]\text{Ran}(R)=\{y|\exists\ x: xRy\}[/math]
Field
The set [math]\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)[/math]
Relation in X
To be a relation in a set [ilmath]X[/ilmath] we must have [math]\text{Field}(R)\subset X[/math]
