Answer
(a) $ S = \{(m,n) : m,n \in \mathbb{Z}\}$
(b) $ T = \{(r,0) , (0,s) : r,s \in \mathbb{R}\}$
Work Step by Step
(a) $S$ is clearly closed under addition , since addition of two integers is again an integer , but it is not closed under scalar multiplication since $ \sqrt2.(1,0)=(\sqrt2,0) \notin S$ .
(b) $T$ is clearly closed under scalar multiplication , but it is not closed under addition since $ (1,0)+(0,1)=(1,1) \notin T $ .
