Verified Solution • 2025–26 Session moderate

QJEEM10Ma98c6: JEE Main 2010 Mathematics - Class 11 JEEM10Ma98c6 Solution: Let S be a non-empty subset of R. Consider the following…

Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P?

Multiple Choice Options

  • A There is no rational number x ∈ S such that x ≤ 0
  • B Every rational number x ∈ S satisfies x ≤ 0
  • C x ∈ S and x ≤ 0 ⇒\Rightarrow x is not rational
  • D There is a rational number x ∈ S such that x ≤ 0

Step-by-Step Verified 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 ⇔\Leftrightarrow ∼\sim P : Every rational number x ∈\in S satisfies x ≤\le 0.

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