Difference between operators and functions 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?

$n$-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.
