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Mathematical Reasoning
easy
Which of the following is the negation of the statement "for all $M\,>\,0$, there exists $x \in S$ such that $\mathrm{x} \geq \mathrm{M}^{\prime \prime} ?$
A
there exists $M\,>\,0$, such that $x \geq M$ for all $x \in S$
B
there exists $M\,>\,0$, there exists $x \in S$ such that $x \geq M$
C
there exists $M\,>\,0$, such that $x < M$ for all $x \in S$
D
there exists $M\,>\,0$, there exists $x \in S$ such that $x < M $
(JEE MAIN-2021)
Solution
$P:$ for all $M\,>\,0$, there exists $x \in S$ such that $x \geq M$
$\sim \mathrm{P}:$ there exists $\mathrm{M}\,>\,0$, for all $\mathrm{x} \in \mathrm{S}$
Such that $x\,<\,M$
Negation of 'there exsits' is 'for all'.
Standard 11
Mathematics
