[Math] Convert universal quantification to existential quantification

Question

I came across following problem

"Every intelligent student is not honest."

And I have to convert this in quantifiers. Straight conversion will be:

∀x [(S(x)∧I(x)) → ¬H(x)] …(i)

However the solution is given in existential quantifier as follows:

∃x [S(x)∧I(x)∧¬H(x)] …(ii)

with explanation "There exist intelligent students who are not honest"

Though this sounds and looks correct, how can I convert (i) to (ii) mathematically I mean without verbal interpretation, may be by double negation?

Answer 1

As per the above comments, the given solution is not correct.

From :

$∀x [(S(x) \land I(x)) \rightarrow \lnot H(x)]$

we have to start using the equivalence between $\forall$ and $\lnot \exists \lnot$ to get

$\lnot \exists x \lnot [(S(x) \land I(x)) \rightarrow \lnot H(x)]$.

Then, we have to apply the tautological equivalence between : $\lnot (p \rightarrow \lnot q)$ and $(p \land q)$ [you can check it with a truth-table] and convert the above formula into :

$\lnot \exists x [S(x) \land I(x) \land H(x)]$.