Question
Let $$S$$ be a non - empty subset of $$R.$$ Consider the following statement :
$$P$$ : There is a rational number $$x \in S$$ such that $$x$$ > 0.
Which of the following statements is the negation of the statement $$P\,?$$
A.
There is no rational number $$x \in S$$ such than $$x \leqslant 0.$$
B.
Every rational number $$x \in S$$ satisfies $$x \leqslant 0.$$
C.
$$x \in S$$ and $$x \leqslant 0\,\,\, \Rightarrow x$$ is not rational.
D.
There is a rational number $$x \in S$$ such that $$x \leqslant 0.$$
Answer :
Every rational number $$x \in S$$ satisfies $$x \leqslant 0.$$
Solution :
$$P$$ : there is a rational number $$x \in S$$ such that $$x$$ > 0
$$ \sim P$$ : Every rational number $$x \in S$$ satisfies $$x \leqslant 0$$