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
 A: For part (a), that's not right - "for all $x$, there is an $x$" should immediately strike you as sounding weird (why is $x$ showing up twice?). You're almost there, though - think a bit more about what you're trying to say.
For (b), let me give a hint: Try to express (b) as a conjunction of two simpler statements:


*

*There is some student in the class without an internet connection.

*It is not the case that there are two students in the class without internet connection.
A: You are correct that "There exists ..." means that there exists at least one. To say that there is exactly one you need to say the following:
$$\exists x(\varphi(x)\land\forall z(\varphi(z)\rightarrow z=x)).$$
Namely, there exists $x$ satisfying whatever, and whenever $z$ satisfies whatever, $z$ has to be equal to $x$. Also note the scope of the existential quantifier is over the entire statement.
(As a mathematical example, There exists a natural number which is larger than $1$; but there exists exactly one natural number which is smaller than $1$ (here we take $0$ to be a natural number))
A: 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))
A: Umm... when I studied math logic back there in my uni, we had exact quantification, expressed as $∃!$
So, let $I(x)$ === "x has internet connection", $P$ === "pupils". Then, a) is represented as
$$
\exists x(x \in P \land \neg I(x))
$$
and b) is represented as
$$
\exists!x(x \in P \land \neg I(x))
$$
Difference between them is only in exactness of second statement: it's not just "not everyone", it's "exactly one" who does not have internet.
We, however, used slightly different notation:
$$
x \in P
$$
$$
\exists ! x : I(x) = false
$$
which, arguably, reads more naturally.
