Answer to Question #2557 in Discrete Mathematics for zaara

Question

Proof that ∀x⇁p(x)≡⇁∃x p(x)
and
Proof that ⇁∀xp(x)≡∃x⇁ p(x)

Answer 1

1)& ∀x⇁p(x)≡⇁∃x p(x)
The left hand side means that “for every x statement p(x) does not hold”, while the right hand side means that “there is no x such that p(x) holds”.

2) LHS means that “not fro every x the statement p(x) holds”, and RHS means that “there exists x such that p(x) does not holds”.

Evidently, in both cases the LHS& statement implies RHS one and vice versa.