Question

Construct a subset of XY plane R^2 that is:

a) closed under vector addition & subtraction, but not under scalar multiplication
b) closed under scalar multiplication, but not vector addition & subtraction

Answer 1

Based on your question the possible answer would be 
a) set of all (u.v) such that u and v are ratios of p/q of integers.
 b) set of all (u.v) where u=0 or v=0.
 How can these vectors be closed for one and not closed for the other operation??