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The codomain should not be Z×Z — firstly, the square root is usually not an integer, but more importantly, there is not a canonical ordering of the pairs you get, especially if we allow negative/complex inputs. The map we really want is

sqrt: C -> (C×C)/~,

where C is the complex numbers and ~ is the equivalence relation generated by (a,b) ~ (b,a) for all a,b in C.

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You're right, I should have said sqrt : N -> R×R. I don't think the order is a problem, we can define the function such that sqrt(x)={a | a²=x}. Sets are inherently unordered, {-2,2} = {2,-2}.

Of course, this is not a surjective function, but it is a function nevertheless.

The issue with sets is that they don’t capture the notion of multiplicity. One would like to say sqrt(0) = {0,0}, but {0,0} is just equal to {0} as a set. The CxC/~ formalism captures the possibility of multiplicity as well as the fact that the output always has size 2.

As an aside, the notation R x R means ordered pairs of reals. So for your definition, you would instead want 2^R as the codomain (i.e. the set of subsets of R).

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