Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)
Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)