The statement is wrong, among other reasons, because the statement
$C(x)$ implies $D(x)$
is true for any $x$ that is not a computer science student, and for any $x$ that takes discrete math. In particular, the statement
There exists $x\in S$ such that $C(x)$ implies $D(x)$. $\qquad\qquad\qquad$ (1)
would be true if nobody is a computer science major. However, I think most people agree that
Some computer science majors take discrete math
should be considered false if there are no computer science majors at all.
Also, the statement (1) would be true if there is at least one person taking discrete math, whether or not that person is a computer science major. So, in a university in which at least one person takes Discrete Math, but no computer science major does, the statement "There exists $x\in S$ such that $C(x)$ implies $D(x)$" would be true, but the statement "Some computer science majors take discrete math" would be false.
What you need to remember is that an implication is true if the antecedent is false, or if the consequent is true. You don't need the antecedent to be true.
What you actually want is:
There exists $x\in S$ such that $C(x)$ and $D(x)$.