I am confused between logical implication and local and. When to use what. Logically i can make sense on which to use but intuitively i am not able to. For ex.,
Not All that glitters is gold
Diagrametric representation: Glitters set is disjoint with gold set
It can be expressed as $\neg \forall x \{ glitters(x) \implies gold(x)\} \equiv \exists x \{ glitters(x) \land \neg gold(x)\}$
But if i think 2 disjoint sets. I think i can write it as $\forall x \{ glitters(x) \land \neg gold(x) \}$ or $\exists x \{ glitters(x) \land \neg gold(x) \}$ or $\forall x \{ \neg glitters(x) \land gold(x) \}$ or $\exists x \{ \neg glitters(x) \land gold(x) \}$ . But if i use logical implication formulae , these seem to be wrong. But the above representations look like 2 disjoint sets right? I am not able to grasp the differences between the above.