JEE Main · 2010 · 11 Apr 2010 · Offline · Mathematical Reasoning *

The statement $P$ claims the existence of something: some rational $x$ in $S$ with $x>0$. Negating an existential claim turns it into a universal one with the inner condition reversed: it is false that such an $x$ exists exactly when every rational $x$ in $S$ fails to be positive. So the negation is that every rational number $x$ in $S$ satisfies $x\le 0$.