Let $S$ be a non-empty subset of $\mathbb{R}$. Consider the following statement: p : There is a rational…
Let $S$ be a non-empty subset of $\mathbb{R}$. Consider the following statement:
p : There is a rational number $x \in \mathrm{~S}$ such that $x\gt0$.
Which of the following statements is the negation of the statement p?
There is a rational number $x \in \mathrm{~S}$ such that $x \leq 0$.
There is no rational number $x \in \mathrm{~S}$ such that $x \leq 0$.
Every rational number $x \in \mathrm{~S}$ satisfies $x \leq 0$.
$x \in \mathrm{~S}$ and $x \leq 0 \Rightarrow x$ is not a rational number.
Solution
$\begin{aligned}
& \text { Given statement is } \\
& \\
& \exists x \in \mathrm{~S} \text {, such that } x\gt0 \\
& \therefore \quad \sim(\exists x \in \mathrm{~S}, \text { such that } x\gt0) \\
& \equiv \forall x \in \mathrm{~S}, x \leq 0
\end{aligned}$
i.e., Every rational number $x \in S$ satisfies $x \leq 0$.