Difference between domain and range for relations and functions?

Question

What is the difference between the definition of domain and range for a relation, and that for a function?

Answer 1

Relations are subsets of products $A\times B$ where $A$ is the domain and $B$ the codomain of the relation.

A function $f$ is a relation with a special property: for each $a\in A$ there is a unique $b\in B$ s.t. $\langle a,b\rangle\in f$.

This unique $b$ is denoted as $f(a)$ and the 'range' of function $f$ is the set $\{f(a)\mid a\in A\}\subseteq B$.

You could also use the notation $\{b\in B\mid\exists a\in A \left[\langle a,b\rangle\in f\right]\}$

Applying that on a relation $R$ it becomes $\{b\in B\mid\exists a\in A \left[\langle a,b\rangle\in R\right]\}$

That set can be labeled as the range of relation $R$.

Also have a look here.

Answer 2

There is no difference. Note, that each function can be seen as a special relation. So $f:A \rightarrow B$ can be seen as a relation $R_f \subseteq A\times B$ such as

$$\forall x \in A: \exists ! y \in B: xR_fy$$

The domain and the range is so defined, such that the relation $R_f$ has the same domain and the range as $f$.

Note: Here I mean with "range" the "image of $f$".