Express statement "not every student in this class will pass the Discrete Mathematics"

Question

I have a bit of a problem with this question:

Express the following statement using predicate function(s), existential or universal quantifier, and/or negation. “not every student in this class will pass the Discrete Mathematics”.

my solution is

-(∀ (( Student(x) && Class(DM,x)) -> Pass(DM,x)) )

is my negative forAll and my && between Student and Class correct?

Thx for any help :)

Answer 1

How about:

$(\exists x\in S)(\sim \operatorname{Pass}(x))$

Where $S$ is the set of students and $\operatorname{Pass}(x)$ is "$x$ passes discrete math".

Answer 2

That is a correct interpretation of the question. As always with English sentences, it's not the only thing you can form. For example, we can form:

$$-(\forall x (DM(x)\rightarrow Pass(x))$$