In order to implement a software I need a function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ such that:
$$(x' \neq x \lor y' \neq y) \Rightarrow f(x,y) \neq f(x',y') $$
It's very easy to implement an algorithm that implements such a function, but I was not able to find an analytical representation of it.
coult you help me?
Answer
The usual "Cantor zig-zag" mapping can be defined as
$$ (x,y) \mapsto \frac{(x+y)(x+y+1)}{2} + x $$
It has the (sometimes) nice property that it is surjective, that is, every natural number is the image of some pair, so nothing goes to waste.
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