MHTCET 23 Apr 2025 Shift 1 Paper

Concept:Negation of a quantified statement swaps $\forall$ and $\exists$, and also negates the inner predicate.Explanation:The given statement is: for all $M > 0$, there exists $x \in S$ such that $x \geq M$.In logical form, it is written as:$$\forall M > 0,\ \exists x \in S \text{ such that } x \geq M.$$The negation of $\forall M > 0$ becomes $\exists M > 0$.The negation of $\exists x \in S$ becomes $\forall x \in S$.The negation of $x \geq M$ becomes $x < M$.Applying these changes to the entire statement gives:$$\exists M > 0 \text{ such that } \forall x \in S,\ x < M.$$In words: there exists some $M > 0$ such that every $x$ in $S$ is less than $M$.Comparing this with the options, it exactly matches Option C.Answer:Option C: $\exists M > 0$ such that $x < M$ for all $x \in S$.