Let S be a non-empty subset of R.

Question

Let S be a non-empty subset of R. Consider the following statement:

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

Which of the following statements is the negation of the statement P?

Accepted Answer

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.

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

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