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).