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 $P$ ?

• A. $x\in S$ and $x\le 0\Rightarrow x$ is not rational.
• B. There is a rational number $x\in S$ such that $x\le 0$
• C. There is no rational number $x\in S$ such that $x\le 0$
• D. Every rational number $x\in S$ satisfies $x\le 0$

Answer 1

The correct option is D Every rational number $x\in S$ satisfies $x\le 0$

$P:$ There is a rational number $x\in S$ such that $x>0$.
Negation of 'there exists' is 'for every'
Negation of '$<$' is '$\ge$'
So, $\sim P:$ Every rational number $x\in S$ satisfies $x\le 0$.