Difference due to scope of quantifiers

I'm trying to understand the difference between the scopes of quantifiers in the following statements and how that impacts taking negations and contrapositives.

Statement 1: $\quad \exists M \in \mathbb{R}^+ \quad \forall x\in \mathbb{R}^+ \quad \big(x\geq M \implies f(x)>0 \big).$

I believe I know how to take the negation and contrapositive of statement 1:

• Negation: $\quad \forall M \in \mathbb{R}^+ \quad \exists x\in \mathbb{R}^+ \quad \big(x\geq M \land f(x)\leq0 \big).$
• Contrapositive: $\quad \exists M \in \mathbb{R}^+ \quad \forall x\in \mathbb{R}^+ \quad \big(f(x)\leq0 \implies x<M \big).$

Statement 2: $\quad \big(\exists M \in \mathbb{R}^+ \quad \forall x\in \mathbb{R}^+ \quad x\geq M\big) \implies f(x)>0 .$

Given statement 2 as above, my questions are:

1. Is there a difference between statement 2 and statement 1, and if so, what is it? An intuitive approach would be appreciated if possible.
2. If there is a difference, how does that impact the negation and contrapositive of statement 2?
3. Finally, if there are any mistakes in my negation and contrapositive of statememt 1, I would appreciate corrections.

Statement 1 doesn't have a contrapositive.

Only conditionals have contrapositives, and 1 isn't a conditional.

More carefully, in a formal context, only formulae whose main logical operator is the conditional (i.e. which are instances of the schema $A \to B$) have contrapositives (the corresponding instance of $\neg B \to \neg A$). And the main operator of 1 is the initial quantifier, so the notion of contraposition doesn't directly apply.

However, if you have a quantified formula $Q_1Q_2\ldots Q_n(A \to B)$ then that will be equivalent to what you get by contraposing $(A \to B)$, i.e. to $Q_1Q_2\ldots Q_n(\neg B \to \neg A)$ but of course leaving the quantifier prefix untouched!

Statement 2 is simply ill-formed in any sensible syntax.