Solve this following

Question:

Which of the following is the negation of the statement "for all $\mathrm{M}>0$, there exists $\mathrm{x} \in \mathrm{S}$ such that $\mathrm{x} \geq \mathrm{M}^{\prime \prime}$ ?

 

  1. there exists $\mathrm{M}>0$, such that $\mathrm{x}<\mathrm{M}$ for all $\mathrm{x} \in \mathrm{S}$

  2. there exists $M>0$, there exists $x \in S$ such that $x \geq M$

  3. there exists $\mathrm{M}>0$, there exists $\mathrm{x} \in \mathrm{S}$ such that $\mathrm{x}<\mathrm{M}$

  4. there exists $M>0$, such that $x \geq M$ for all $x \in S$


Correct Option: 1

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

Negation of 'there exsits' is 'for all'.

 

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