Answer
Logical Equivalences:
p→q≡¬p∨q
De Morgan's Law for Qualifiers:
$$\neg \exists x P(x) \equiv \forall x \neg P(x) $$$$\neg \forall x P(x) \equiv \exists x \neg P(x) $$
a) $\forall x P(x) \rightarrow A$ is logically equivalent with $\neg(\forall xP(x)) \lor A $ by above Logical Equivalence.
Use De Morgan's Law for Qualifiers:
$$\equiv \exists x \neg P(x) \lor A$$.
Use result of exercise 46 and Logical Equivalence
$$\equiv \exists x (\neg P(x) \lor A)$$.
$$\equiv \exists x (P(x) \rightarrow A)$$.
Thus, $\forall x P(x) \rightarrow A$ is logically equivalent with $\exists x (P(x) \rightarrow A)$
b) Use Logical Equivalence,
$\exists xP(x) \rightarrow A \equiv \neg \exists xP(x) \lor A$
Use De Morgan's Law for Qualifiers:
$$\equiv \forall x (\neg P(x)) \lor A$$.
By previous exercise 46, this is logically equivalent to
$$\equiv \forall x (\neg P(x) \lor A)$$.
$$\equiv \forall x (P(x) \rightarrow A)$$.
Thus, $\exists x P(x) \rightarrow A$ is logically equivalent with $\forall x (P(x) \rightarrow A)$
Work Step by Step
See answer
