Q81 / 150 · MHT CET 2025 (23 Apr Shift 1) · Mathematics

Which of the following is the negation of the statement " For all $\mathrm{M}>0$, there exist $x \in s$ such that $x \geqslant \mathrm{M}$ "

  • A
    $\exists \mathrm{M}>0$ such that $x \geqslant \mathrm{M}$ for all $x \in \mathrm{~s}$
  • B
    $\exists \mathrm{M}>0, \exists x \in \mathrm{~s}$ such that $x \geqslant \mathrm{M}$
  • Correct answer
    C
    $\exists \mathrm{M}>0$ such that $x < \mathrm{M}$ for all $x \in \mathrm{~s}$
  • D
    $\exists \mathrm{M}>0$, there exist $x \in \mathrm{~s}$ such that $x < \mathrm{M}$

Solution

The statement $"$For all $M>0$, there exists $x \in s$ such that $x \geqslant M$"$ has logical structure $\forall M>0, \exists x \in s : x \geqslant M$.</p><p>Negating quantifiers transforms $\forall$ to $\exists$ and vice versa, while the inequality $x \geqslant M$ becomes $x < M$.</p><p>The negation is therefore $\exists M>0, \forall x \in s : x < M$.</p><p>This corresponds to option <strong>C</strong>: $\exists M>0$ such that $x < M$ for all $x \in s$.

Hence, the correct answer is option: 'C'.

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