Relation
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]
Functions, equivalence relations and orderings are special kinds of relation
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
Proof that domain, range and field exist may be found here
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]
Properties of relations
Symmetric
A relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] is symmetric if for all [ilmath]a,b\in A[/ilmath] we have that [ilmath]aRb\implies bRa[/ilmath] - a property of equivalence relations
Antisymmetric
A binary relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] is antisymmetric if for all [ilmath]a,b\in A[/ilmath] we have [math]aRb\text{ and }bRA\implies a=b[/math]
Symmetric implies elements are related to each other, antisymmetric implies only the same things are related to each other.
Reflexive
For a relation [ilmath]R[/ilmath] and for all [ilmath]a\in A[/ilmath] we have [ilmath]aRa[/ilmath] - [ilmath]a[/ilmath] is related to itself.
Transitive
A relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] is transitive if for all [ilmath]a,b,c\in A[/ilmath] we have [math][aRb\text{ and }bRc\implies aRc][/math]
Asymmetric
A relation [ilmath]S[/ilmath] in [ilmath]A[/ilmath] is asymmetric if [ilmath]aSb\implies(b,a)\notin S[/ilmath], for example [ilmath]<[/ilmath] has this property, we can have [ilmath]a<b[/ilmath] or [ilmath]b<a[/ilmath] but not both.
