How to convert these quantified statements into Prenex Normal Form (PNF)?
(a) $\exists x P(x) \, \lor \, \exists x Q(x) \, \lor \, A$, where $A$ is a proposition not involving any quantifiers.
My Attempt:
We find that $$ \begin{align} & \qquad \exists x P(x) \, \lor \, \exists x Q(x) \, \lor \, A \\ &\equiv \exists x \big( P(x) \lor Q(x) \big) \lor A \\ & \qquad \qquad \mbox{[ using Exercise 47, Sec. 1.4 ]} \\ &\equiv \exists x \big( P(x) \lor Q(x) \lor A \big). \\ & \qquad \qquad \mbox{[ using Exercise 48 (b), Sec. 1.4 ]} \end{align} $$
Am I right?
(b) $\lnot \big(\forall x P(x) \, \lor \, \forall x Q(x) \big)$.
My Attempt:
We find that $$ \begin{align} & \qquad \lnot \big(\forall x P(x) \, \lor \, \forall x Q(x) \big) \\ &\equiv \big( \lnot \forall x P(x) \big) \land \big( \lnot \forall x P(x) \big) \\ & \qquad \mbox{[ using 2nd DeMorgan's law in } \\ &\qquad \qquad \mbox{ Table 6, Sec. 1.3 ]} \\ &\equiv \big( \exists x \lnot P(x) \big) \land \big( \exists x \lnot Q(x) \big) \\ & \qquad \mbox{[ using 2nd DeMorgan's law for quantifiers in } \\ &\qquad \qquad \mbox{ Table 2, Sec. 1.4 ]} \\ &\equiv \exists x \exists y \big( \lnot P(x) \, \land \, \lnot Q(x) \big) \\ &\qquad \mbox{[ using a result analogous to that } \\ &\qquad \qquad \mbox{ in Exercise 48, Sec. 1.5 ]}. \end{align} $$
Am I right?
(c) $\exists x P(x) \, \rightarrow \, \exists x Q(x)$.
My Attempt:
We note that $$ \begin{align} & \qquad \exists x P(x) \, \rightarrow \, \exists x Q(x) \\ & \equiv \lnot \big( \exists x P(x) \big) \, \lor \, \big( \exists x Q(x) \big) \\ & \qquad \mbox{[ using the conditional-disjunction equivalence ]} \\ &\equiv \big( \forall x \lnot P(x) \big) \, \lor \, \big( \exists x Q(x) \big) \\ &\equiv \forall x \exists y \big( \lnot P(x) \, \lor \, Q(y) \big) \\ &\qquad \mbox{[ using the result of Exercise 49 (b), Sec. 1.5 ]}. \end{align} $$
Am I right?
Are my solutions to all of these three parts correct and clear enough? Or, are there any issues?