Difference between operators and functions

Question

Binary operators are maps on a set into itself, for example $* : R \rightarrow R$. There are also unary and trinary operators.

What is the difference between, say a trinary operator of three arguments, and a function of three arguments?

Answer 1

$n$-ary operations are special kinds of functions with domains of the form $A^n$ and codomains of the form $A$ for some set $A$. They are functions, but not all functions are $n$-ary operations.

What you say in your question isn't quite right:

A unary operation is a function $\ast\colon A\to A$ as it takes one input.

A binary operation is a function $\ast\colon A\times A\to A$ as it takes two inputs.

A ternary operation is a function $\ast\colon A\times A\times A\to A$ as it takes three inputs.

Answer 2

First of all, a binary operator takes two arguments (hence binary), so you want $\ast : R\times R \to R$.

The difference between a ternary operator on $R$ and a function of three arguments on $R$ is the codomain; the former has codomain $R$ (i.e. $R\times R\times R \to R$), whereas the latter could have a codomain other than $R$ (i.e. $R\times R\times R \to S$).