Difference between revisions of "Relation"
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==Notation== | ==Notation== | ||
Rather than writing {{M|(x,y)\in R}} to say {{M|x}} and {{M|y}} are related we can instead say {{M|xRy}} | Rather than writing {{M|(x,y)\in R}} to say {{M|x}} and {{M|y}} are related we can instead say {{M|xRy}} | ||
| − | + | ==Basic terms== | |
| − | ==Domain== | + | ===Domain=== |
The set of all {{M|x}} which are related by {{M|R}} to some {{M|y}} is the domain. | The set of all {{M|x}} which are related by {{M|R}} to some {{M|y}} is the domain. | ||
<math>\text{Dom}(R)=\{x|\exists\ y: xRy\}</math> | <math>\text{Dom}(R)=\{x|\exists\ y: xRy\}</math> | ||
| − | ==Range== | + | ===Range=== |
The set of all {{M|y}} which are a relation of some {{M|x}} by {{M|R}} is the range. | The set of all {{M|y}} which are a relation of some {{M|x}} by {{M|R}} is the range. | ||
<math>\text{Ran}(R)=\{y|\exists\ x: xRy\}</math> | <math>\text{Ran}(R)=\{y|\exists\ x: xRy\}</math> | ||
| − | ==Field== | + | ===Field=== |
The set <math>\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)</math> | The set <math>\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)</math> | ||
| − | ==Relation in X== | + | ===Relation in X=== |
To be a relation in a set {{M|X}} we must have <math>\text{Field}(R)\subset X</math> | To be a relation in a set {{M|X}} we must have <math>\text{Field}(R)\subset X</math> | ||
| − | ==Image of A under R== | + | ==Images of sets== |
| + | ===Image of A under R=== | ||
This is just the set of things that are related to things in A, denoted <math>R[A]</math> | This is just the set of things that are related to things in A, denoted <math>R[A]</math> | ||
<math>R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}</math> | <math>R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}</math> | ||
| − | ==Inverse image of B under R== | + | ===Inverse image of B under R=== |
As you'd expect this is the things that are related to things in B, denoted <math>R^{-1}[B]</math> | As you'd expect this is the things that are related to things in B, denoted <math>R^{-1}[B]</math> | ||
<math>R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}</math> | <math>R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}</math> | ||
| + | |||
| + | ===Important lemma=== | ||
| + | It is very important to know that the inverse image of B under R is the same as the image under <math>R^{-1}</math> | ||
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
Revision as of 22:12, 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]
Basic terms
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]
Images of sets
Image of A under R
This is just the set of things that are related to things in A, denoted [math]R[A][/math]
[math]R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}[/math]
Inverse image of B under R
As you'd expect this is the things that are related to things in B, denoted [math]R^{-1}[B][/math]
[math]R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}[/math]
Important lemma
It is very important to know that the inverse image of B under R is the same as the image under [math]R^{-1}[/math]
