Contrapositive, Converse and Inverse of statements with multiple quantifiers

Question

My textbook only touched on negation of statements with multiple quantifiers, and I would like to know:

For a statement like

$\forall M>0, \exists \delta > 0$ such that if $0 < |x-a| < \delta$ then $|f(x)| > M,$

is its contrapositive

$\forall M>0, \exists \delta > 0$ such that if ~$(|f(x)| > M)$ then ~$(0 < |x-a| < \delta)\quad?$
Do the converse and inverse similarly just affect the if-else?

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The contrapositive of $P \to Q$ is $\lnot Q \to \lnot P$ and the two are equivalent. Thus, you can exchange them inside the quantifiers and the result is equiv to the original formula. But this is not what you get when you negate the full formula with quantifiers. — Mauro ALLEGRANZA, Feb 02 '23 at 08:24
The converse of $P \to Q$ is $Q \to P$ and it is not equivalent to the original one. Thus, you cannot exchange them without affecting the meaning and truth value of the formula. — Mauro ALLEGRANZA, Feb 02 '23 at 08:26

ryang · Accepted Answer · 2023-02-11T14:30:47.967

Yes, Contrapositive, Converse and Inverse all refer and apply to conditional formulae and affect only their antecedents and consequents and not any surrounding quantifier or connective.

This is not to say that they never alter quantifiers: the contrapositive of $$\forall x\big(Px\to\exists y Qxy\big)$$ is $$∀x(∀y¬Qxy→¬Px).$$ (Strictly speaking, we are referring to the contrapositive of the string within the parentheses.)

How can I know when to negate quantifiers when taking the contrapositive of a statement?

Contrapositive, Converse and Inverse of statements with multiple quantifiers

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