JEE Main 2010 — Mathematical Reasoning Question with Solution

Given that S is a non-empty subset of R.

$$\bullet$$ P : There is a rational number x $$\in$$ S such that x > 0.

Now, we need to find the negation of P. Clearly, P is equivalent to saying that "There is a positive rational number in S.

So, its negation ($$\sim$$ P) is "There is no positive rational number in S".

Thus, for $$\sim$$ P : There exists no positive rational number in S.

$$\bullet$$ $$\iff$$ $$\sim$$ P : Every rational number x $$\in$$ S satisfies x $$\le$$ 0.