Mathematical Difference between "there is one" and "there is EXACTLY one" I know that I can say ∃x(P(x)) which means there is at least one x for P(x), but how do I express for exactly one?
Here's the questions:
(a) Not everyone in your class has an internet connection.
(b) Everyone except one student in your class has an internet connection.
So for the first one I wrote:
(a) ∀x∃x(¬I(x))
"For all x there exists an x (or more) such that an x does not have an internet connection" (where I is the state of having an internet connection)
(b) Don't know how to express
I could be wrong please correct me since i'm pretty new to expressing this all mathematically
Thanks for help

Expressing "exactly one" close to natural language:
There is someone (∃x) without Internet (¬I(x)) and (∧) everyone else (∀y (y≠x→..) has Internet (I(y)): 
∃x(¬I(x)∧∀y(y≠x→I(y)))
With bounded quantifiers the same could be expressed a little simpler, and incorporating "in your class" clauses.
There is someone in your class (∃(x∈C)) without Internet (¬I(x)) and (∧) everyone else in your class (∀(y∈C\{x})) has Internet (I(y)): 
∃(x∈C)((¬I(x)∧∀(y∈C\{x})I(y))

(a) Not everyone in your class has an internet connection.

∃x(¬I(x))

b) Everyone except one student in your class has an internet
  connection.

∃y∀z(y≠z→I(z))
