Question

Let $S$ be a non-empty subset of $R$ Consider the following statement:
$P:$ There exist's 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 $x\le 0.$