I have a longstanding issue about how to construct contrapositives that I can't seem to resolve on my own. In general, if $Q \rightarrow P$ is true, then so is $\neg P \rightarrow \neg Q$. What I often find confusing when constructing the contrapositive is deciding upon the terms that will receive the negation. For example, consider this simple analysis theorem:

If $A\_{1}, A\_{2},\ldots, A\_{m}$ are countable sets, then the union $A\_{1}\cup A\_{2}\cup \ldots \cup A\_{m}$ is countable. 

The form of the contrapositive is 

$\neg(A\_{1}\cup A\_{2}\cup \ldots \cup A\_{m}$ countable)$\rightarrow \neg(A\_{1}, A\_{2},\ldots, A\_{m}$ are countable sets).

In this case, does the negation of the premise in the contrapositive apply to the union operators or just to the countability property? And does the negation of the conclusion apply to what seems to be an implied universal quantifier on the $A\_{i}$'s? In other words, is the contrapositive

$A\_{1} \cap A\_{2}\cap \ldots \cap A\_{n}$ uncountable $\rightarrow$ there exists $A\_{i} \in \{{A\_{1}, A\_{2},\ldots, A\_{m}\}}$ that is uncountable 

or

$A\_{1} \cap A\_{2}\cap \ldots \cap A\_{n}$ uncountable $\rightarrow A\_{1}, A\_{2},\ldots, A\_{m}$ are uncountable sets

or just

$A\_{1}\cup A\_{2}\cup \ldots \cup A\_{m}$ uncountable$\rightarrow A\_{1}, A\_{2},\ldots, A\_{m}$ are uncountable sets.

Similarly, for the theorem

If $A\_{n}$ is a countable set for each $n \in \mathbb{N}$, then $\cup\_{n=1}^{\infty} A\_{n}$ is countable. 

Is the contrapositive

$\cup\_{n=1}^{\infty} A\_{n}$ is uncountable $\rightarrow A\_{n}$ is an uncountable set for each $A\_{n}$

or 

$\cup\_{n=1}^{\infty} A\_{n}$ is uncountable $\rightarrow$ there exists some $n \in \mathbb{N}$ such that $A\_{n}$ is uncountable

or

$\cap\_{n=1}^{\infty} A\_{n}$ is uncountable $\rightarrow$ there exists some $n \in \mathbb{N}$ such that $A\_{n}$ is uncountable

I hope these illustrations give some idea of the basis of my confusion (my apologies for the lame formatting). Any clarifying remarks will be greatly appreciated!