Question

If $S$ 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 statement $P$$?$

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

Answer 1

The correct option is C

Every rational number $x\in S$ such that $x\le 0$

Finding the negation of statement $P$:

As per the given statement $P$, there is a rational number $x\in S$, so the negation of this part will be every rational number $x\in S$.

Let us focus on the second part that says $x>0$, so the negation of this will be $x\le 0$.

Therefore, the negation of $P$ or $~P$ will be every rational number $x\in S$ such that $x\le 0$..

Hence, the correct option is (C).