How are injections, surjections and bijections described for functions with an ordered pair as an argument?

Question

When given a function with an ordered pair as argument i.e.

Z × Z → Z given by f(m, n) = m − n − 1

How should injections, surjections and bijections be described? Should there be a domain of tuples ie. (1,1) (1,2) (2,1) (2,2)? Or possibly an extra domain or codomain?

Any help would be greatly appreciated.

Answer 1

Those aren't functions with multiple arguments. Those are functions taking a single argument in the domain $\mathbb{Z}\times\mathbb{Z}$. The definitions are exactly as for any other function.