# Let S be a non-empty subset of R. Consider the following statement:P : There is a rational number $\dpi{100} x\in S$ such that $\dpi{100} x> 0.$Which of the following statements is the negation of the statement P ? Option 1) There is a rational number $x\in S$ such that $x\leq 0$.    Option 2) There is no rational number $x\in S$ such that $x\leq 0$. Option 3) Every rational number $x\in S$ satisfies  $x\leq 0$. Option 4) $x\in S$ and $x\leq 0$ $\Rightarrow$ $x$ is not rational.

D Divya Saini

As we learnt in

Negation of a Statement -

Symbol of negation is $\sim$

- wherein

We write it up as $\sim$p

$P:$ There is rational numbers  $x\epsilon S$  such that $x> 0.$

negation is every rational number $x\epsilon S$  satisfies $x\leq 0.$

Since negation of $x> 0$ is $x\leq 0 .$

Option 1)

There is a rational number $x\in S$ such that $x\leq 0$.

Incorrect option

Option 2)

There is no rational number $x\in S$ such that $x\leq 0$.

Incorrect option

Option 3)

Every rational number $x\in S$ satisfies  $x\leq 0$.

Correct option

Option 4)

$x\in S$ and $x\leq 0$ $\Rightarrow$ $x$ is not rational.

Incorrect option

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