Question

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

• A. There is no rational number $x\in S$ such that $x\le 0$
• B. For every rational number $x\in S,$ $x\le 0$
• C. For every rational number $x\in S,$ $x>0$
• D. There exists a rational number $x\in S$ such that $x\le 0$

Answer 1

The correct option is B For every rational number $x\in S,$ $x\le 0$

$P:$ At least one rational number $x\in S$ such that $x>0$
$\sim P:$ There is no rational number $x\in S$ such that $x>0$
i.e., All rational numbers $x\in S$ satisfy $x\le 0.$