Let S be a set consisting of 31 positive real numbers. For each non-empty subset A ⊂ S let
f(A) be the product of all elements of A. We say that a subset A ⊂ S is rational if f(A) is
a rational number. We say that a subset A ⊂ S is irrational if f(A) is an irrational number.
Is there any set S having exactly 2023 rational subsets? Is there any set S having exactly
2025 irrational subsets?