Here are the two counterexamples for reference.
Universal Quantifier
Let us consider the following English statement and try to convert it into logical statement using quantifier and predicates
All users of GateOverflow have studied calculus and are helpful.
If the domain of the statement is GateOverflow webstite then we have the following logical interpretation -
$$\forall x \ H(x) \wedge C(x)$$
where, $H(x) = x \ is \ a \ helpful \ user$ and $C(x) = x \ has \ studied \ Calculus$.
Existential Quantifier
Let us consider another similar english statement
There is a user in GateOverflow who has studied calculus only if he is helpful
Now, it is implicit from the English interpretation that we have no information about the helpful nature of those users who have not studied calculus. Therefore, even if just one of the users of GateOverflow has studied calculus then that person is helpful.
Using the domain and predicates from previous example we have
$$\exists x \ C(x) \rightarrow H(x)$$