Find the cardinality of the set $\{ m + \sqrt 5 n + \sqrt3 p : m, n, p \in\mathbb Q\}.$

Question

I'm trying to find the cardinality of this set:

$$\{ m + \sqrt 5 n + \sqrt3 p : m, n, p \in\mathbb Q\}.$$

However, this is somewhat a tougher set to calculate a cardinality for. How do we go about calculating the cardinality for this set?

Answer 1

Let $S$ be the set in question ($S = \{m + n \sqrt{3} + p \sqrt{5} : m, n, p \in \mathbb{Q}\}$).

Clearly, there is a surjection $s : \mathbb{Q}^3 \to S$ defined by $s(m, n, p) = m + n \sqrt{3} + p \sqrt{5}$. Since $S$ is infinite and $\mathbb{Q}^3$ is countable, it follows that $S$ is countable.