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Let $S$ be a non empty subset of $R$. Consider the following statement: $P$: There is a rational number $x \in S$ such that $x\gt 0$. Then, which of the following statements is the negation of statement $P$:

$\begin{array}{1 1}(A) x \in \; and x \leq 0 \Rightarrow \text{x is not rational} \\ (B) \text{Every rational number} x \in \; satisfies\; x \leq 0 \\(C) \text{There is no rational number} x \in such that x \leq 0 \\(D) \text{There is a rational number} x \in such\; that \;x \leq 0 \end{array}$

1 Answer

The given statement is $P$: There is a rational number $x \in S$ such that $x\gt 0$.
The negation would be: There is no rational number $x \in S$ such that $x \gt 0$ which is equivalent to all rational numbers $x \in S$ satisfy $x \leq 0$
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