Which of the following is the negation of the statement "for all $$M \gt 0$$, there exists $$x \in S$$ such that $$x \geq M$$"?

We start with the given statement:

$$\text{(I)}\;:\;\forall\,M \gt 0,\;\exists\,x\in S\text{ such that }x\ge M.$$

The words “for all” correspond to the universal quantifier $$\forall$$ and the words “there exists” correspond to the existential quantifier $$\exists$$. To find the negation of a quantified statement, we use the standard logical rules:

1. The negation of $$\forall$$ (for all) is $$\exists$$ (there exists).
2. The negation of $$\exists$$ (there exists) is $$\forall$$ (for all).
3. Inside, we must also negate the inner predicate.

Applying rule 1 to statement (I) we replace the leading $$\forall$$ by $$\exists$$. So we write:

$$\neg\text{(I)}:\;\exists\,M \gt 0\ \bigl[\neg\,(\exists\,x\in S\text{ such that }x\ge M)\bigr].$$

Now we focus on the inner part $$\exists\,x\in S\text{ such that }x\ge M$$ and apply rule 2: its negation changes $$\exists$$ to $$\forall$$ and, according to rule 3, the condition $$x\ge M$$ switches to its opposite $$x\lt M$$. Thus,

$$\neg\bigl(\exists\,x\in S\text{ such that }x\ge M\bigr)\;=\;\forall\,x\in S,\;x\lt M.$$

Substituting this negation back, we get

$$\neg\text{(I)}:\;\exists\,M \gt 0\text{ such that }\forall\,x\in S,\;x\lt M.$$

Reading this sentence in plain words: “There exists a positive number $$M$$ such that every element $$x$$ of the set $$S$$ is strictly less than $$M$$.”

Now we compare this with the options given:

A. There exists $$M \gt 0$$ such that $$x \lt M$$ for all $$x \in S$$.
B. There exists $$M \gt 0$$, there exists $$x \in S$$ such that $$x \ge M$$.
C. There exists $$M \gt 0$$, there exists $$x \in S$$ such that $$x \lt M$$.
D. There exists $$M \gt 0$$ such that $$x \ge M$$ for all $$x \in S$$.

Option A reproduces exactly the negated statement we have derived. The other options either keep both quantifiers as existential (B, C) or invert the inequality without changing both quantifiers appropriately (D). Therefore Option A is the only correct negation.

Hence, the correct answer is Option A.