Answer to Question #2557 in Discrete Mathematics for zaara
Proof that ∀x⇁p(x)≡⇁∃x p(x)
Proof that ⇁∀xp(x)≡∃x⇁ p(x)
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.