What case and reason to use implication and conjunction?

Question

I must honestly to say that I cannot understand what and where, how to use the quantifier when its combined with logic connective, specifically implication and conjunction

In the someone's lecture, they have the problem:

"If a user is active, at least one network link will be available."

So let:

A(u) represent “User u is active.”

S(n, x) represent “Network link n is state x"

The result is: ∃u A(u) → ∃n S(n,available) , although I almost see ∃ is usually come with the "∧" conjunctions.So, I was think like Why can not it be: ∃u A(u) ∧ ∃n S(n,available)

In short, is that "the quantifier" is not mostly decided "What the logic connective" can be? and We have to based on the context to decide the logic connective too ?Thanks all

Answer 1

The result is: ∃u A(u) → ∃n S(n, available) , although I almost see ∃ is usually come with the "∧" conjunctions.

True, but in this case, the '→' is outside of the scope of the first '∃'. Shown with implicit bracketing:

∃u [ A(u) ] → ∃n [ S(n, available) ]

So, I was think like Why can not it be: ∃u A(u) ∧ ∃n S(n, available)

That is not a conditional statement. It states, in effect, that user u is active and that at least one link is active.