Linear Algebra – Example of a Subset of $\mathbb{R}^2$ Closed Under Vector Addition but Not Scalar Multiplication

Question

I've found several examples which are closed under scalar multiplication, but not vector addition, but I can't come up with one that is closed under vector addition, but not scalar multiplication.

Answer 1

$S=\{(r,s)\in\mathbb{R}^2:r,s\in\mathbb{Q}\}$ works as well.