$S(x)$ means $x$ is a student.
$C(x)$ means $x$ drinks coffee.
I need to turn the statement
"It is not true that all the students drink coffee."
into predicate logic. This is what I have.
¬(∀x, ¬C(x) ∨S(x))
I think it is somewhat correct but I am not sure. I think I am wrong because the next part asks me to simplify the logical expression using the logical equivalence of $p \implies q \equiv \neg p \vee q$.
What is that asking me to do? Do I have to use my current answer and convert it or is this a completely new thing?
Following from the comments, we have $S(x) = $ "$x$ is a student", and $C(x) = $ "$x$ drinks coffee."
So for every student to drink coffee means: $\forall x(S(x) \to C(x))$. We have the statement that not all students drink coffee, so we will negate our statement. First lets convert $\to$ to its logical equivalent: $p \to q \equiv \lnot p \lor q$.
Now let's throw the negation in front of our statement and converting $\to$ to its logical equivalent, we have: $$\lnot \forall x(\lnot S(x) \lor (C(x)) \\ \exists x \lnot (\lnot S(x) \lor C(x)) \qquad \lnot \forall x \equiv \exists x \\ \text{Now we will follow the negation through.} \\ \exists x(\lnot \lnot S(x) \land \lnot C(x)) \qquad \lnot \lor \equiv \land \\ \boxed{\exists x(S(x) \land \lnot C(x))} \qquad \lnot \lnot p\equiv p \space \text{Double negation.} $$
Now this translates to, "There exists a student and they do not drink coffee."