The binary operation ◻ is defined as follows:

Which one of the following is equivalence to $$P \vee Q$$?

Which of the following is the negation of $$$\left[ {\forall x,\alpha \to \left( {\exists y,\beta \to \left( {\forall u,\exists v,\gamma } \right)} \right)} \right]?$$$

$$P$$ and $$Q$$ are two propositions. Which of the following logical expressions are equivalent?

$${\rm I}.$$ $${\rm P}\, \vee \sim Q$$
$${\rm I}{\rm I}.$$ $$ \sim \left( { \sim {\rm P} \wedge Q} \right)$$
$${\rm I}{\rm I}{\rm I}.$$ $$\left( {{\rm P} \wedge Q} \right) \vee \left( {{\rm P} \wedge \sim Q} \right) \vee \left( { \sim {\rm P} \wedge \sim Q} \right)$$
$${\rm I}V.$$ $$\left( {{\rm P} \wedge Q} \right) \vee \left( {{\rm P} \wedge \sim Q} \right) \vee \left( { \sim {\rm P} \wedge Q} \right)$$

Let fsa and $$pda$$ be two predicates such that fsa$$(x)$$ means $$x$$ is a finite state automation, and pda$$(y)$$ means that $$y$$ is a pushdown automation. Let $$equivalent$$ be another predicate such that $$equivalent$$$$(a,b)$$ means $$a$$ and $$b$$ are equivalent. Which of the following first order logic statements represents the following:

Each finite state automation has an equivalent pushdown automation.