The binary operation ◻ is defined as follows:

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

Consider the following well-formed formulae:

$${\rm I}.$$ $$\,\,\neg \forall x\left( {P\left( x \right)} \right)$$
$${\rm I}{\rm I}.\,\,\,\,\,\,\neg \exists x\left( {P\left( x \right)} \right)$$
$${\rm I}{\rm I}{\rm I}.\,\,\,\,\,\,\neg \exists x\left( {\neg P\left( x \right)} \right)$$
$${\rm I}V.\,\,\,\,\,\,\exists x\left( {\neg P\left( x \right)} \right)$$

Which of the above are equivalent?

Which one of the following is the most appropriate logical formula to represent the statement:

"$$Gold\,and\,silver\,ornaments\,are\,precious$$"

The following notations are used:
$$G\left( x \right):\,\,x$$ is a gold ornament.
$$S\left( x \right):\,\,x$$ is a silver ornament.
$$P\left( x \right):\,\,x$$ is precious.

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.