The theorem states. Let A be a set and Let c be a non empty countable set disjoint from A. If A has a denumerable subset then A union C is equinumerous to A and conversely.
I was thinking of letting A be the rational numbers, and letting C be the irrational numbers that way it's disjoint, and then the subset of A would be integers, but then so the union of integers and irrational numbers would be equinumerous to rational numbers, but that doesn't help with the equinumerous of irrational and real numbers. Not sure what to do.