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

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.
