Question #2557

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

Expert's answer

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.

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.

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