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]

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]