Difference between revisions of "Relation"
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(Created page with "A set {{M|R}} is a binary relation if all elements of {{M|R}} are ordered pairs. That is for any {{M|z\in R\ \exists x\text{ and }y:(x,y)}} ==Notation== Rat...") |
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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== | ||
| + | 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> | ||
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
Revision as of 22:07, 1 March 2015
A set R is a binary relation if all elements of R are ordered pairs. That is for any z∈R ∃x and y:(x,y)
Contents
[hide]Notation
Rather than writing (x,y)∈R to say x and y are related we can instead say xRy
Domain
The set of all x which are related by R to some y is the domain.
Dom(R)={x|∃ y:xRy}
Range
The set of all y which are a relation of some x by R is the range.
Ran(R)={y|∃ x:xRy}
Field
The set Dom(R)∪Ran(R)=Field(R)
Relation in X
To be a relation in a set X we must have Field(R)⊂X
Image of A under R
This is just the set of things that are related to things in A, denoted R[A]
R[A]={y∈Ran(R)|∃x∈A:xRa}
Inverse image of B under R
As you'd expect this is the things that are related to things in B, denoted R−1[B]
R−1[B]={x∈Dom(R)|∃y∈B:xRy}
