Let $S$ be a non-empty subset of R. Consider the following statement: $\mathrm{P}$ : There is a rational…

Let $S$ be a non-empty subset of R. Consider the following statement: $\mathrm{P}$ : There is a rational number $\mathrm{x} \in \mathrm{S}$ such that $\mathrm{x}>0$. Which of the following statements is the negation of the statement $P$ ?
  1. There is no rational number $x \in S$ such that $x \leq 0$
  2. Every rational number $x \in S$ satisfies $x \leq 0$
  3. $x \in S$ and $x \leq 0 \Rightarrow x$ is not rational
  4. There is a rational number $x \in S$ such that $x \leq 0$

Solution

$P$ : there is a rational number $x \in S$ such that $x>0$ $\sim P$ : Every rational number $\mathrm{x} \in \mathrm{S}$ satisfies $\mathrm{x} \leq 0$

Asked in: JEE Main 2010

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