Relation

A set $R$ is a binary relation if all elements of $R$ are ordered pairs. That is for any $z\in R\ \exists x\text{ and }y:(x,y)$

Notation

Rather than writing $(x,y)\in R$ 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.

$$\text{Dom}(R)=\{x|\exists\ y: xRy\}$$

Range

The set of all $y$ which are a relation of some $x$ by $R$ is the range.

$$\text{Ran}(R)=\{y|\exists\ x: xRy\}$$

Field

The set $$\text{Dom}(R)\cup\text{Ran}(R)=\text{Field}(R)$$

Relation in X

To be a relation in a set $X$ we must have $$\text{Field}(R)\subset X$$

Image of A under R

This is just the set of things that are related to things in A, denoted $$R[A]$$

$$R[A]=\{y\in\text{Ran}(R)|\exists x\in A:xRa\}$$

Inverse image of B under R

As you'd expect this is the things that are related to things in B, denoted $$R^{-1}[B]$$

$$R^{-1}[B]=\{x\in\text{Dom}(R)|\exists y\in B:xRy\}$$