Minkowski Set Addition

Question

What is the Minkowski sum of Q to itself, where Q is the set of all rational numbers?

I can't find a way to solve this with freshman knowledge.

Thank you in advance.

Answer 1

Wouldn't it be $\mathbb{Q}$ since $$\frac{m}{n}+\frac{m'}{n'}$$ is rational for all nonzero integers $n,n'$ and all integers $m,m'$?

Answer 2

The sum of tow rationals is rational, so $$\mathbb{Q}+\mathbb{Q}\subseteq \mathbb{Q}$$ To show the opposite containment, we need to show that any rational can be expressed as the sum of two rationals. Take one of the two rationals to be $0.$