Georges (x) means "Georges knows x" Math (x) means "x is a math student"
I can only use those symbols : ∧, ∨, ¬, ∀, ∃, →, ↔
1) Georges knows some student who isn't a Math student
2) The only students known to Georges are Math students
3) Georges knows exactly one student.( I can also use = and ≠)
for 1 : ∃(x), Georges(x) → ¬ Math (x) for 2 : ∃(x), Georges(x) → Math (x) for 3 : ∃(x,y), Georges(x ≠ y) → (¬∀x ∧ ∃y)
If you can help me, I spent a lot of time on it, thanks!
To simplify the statements some, I will refer to $\text{Georges}(x)$ instead as $G(x)$ and $\text{Math}(x)$ instead as $M(x)$.
(1): "There is some student who is not a math student that George knows"
Reworded, "There is some student such that George knows them and such that they are not a math student"
$\exists x, G(x)\wedge M(x)$
(2): "The only students known to George are math students."
Reworded, "*For all students, if George knows them then they are a math student."
$\forall x (G(x)\implies M(x))$
Equivalently, "There does not exist a student that George knows that is not a math student"
$\not\exists x,G(x)\wedge \neg M(x)$
(3): "George knows exactly one student"
This can be written a number of ways...
"There is a unique student such that George knows them"
$\exists ! x, G(x)~~~~~~$ This is the easiest way to write it, however the notation is not universally known and was not explicitly included in the list of symbols you are allowed to use.
"There is at least one student such that George knows them and if two unique students are chosen, at least one of them George does not know"
$(\exists x, G(x))\wedge (\forall (x,y), x\neq y\implies (\neg(G(x))\vee \neg(G(y))))$
"There is at least one student such that George knows them and if two (not necessarily distinct) students are chosen and George knows both of them, then they were in fact the same student"
$(\exists x, G(x))\wedge (\forall(x,y), (G(x)\wedge G(y))\implies x=y)$
These final two should look quite similar, and the astute observer would notice that the implications at the end are equivalent, one merely being the contrapositive of the other.