How do I solve this discrete math/quantifier problem?

Question

So I have to "Use equivalences to rewrite the propositions into a logically equivalent proposition in which quantifiers only appear directly in front of predicate symbols (only one quantifier can appear directly in front of the predicate symbol". Here is the problem: $(\forall x)(P(x) \land (Q(b) \lor R(x))$. I have to solve several problems like this one, but I just don't understand where to start. I know we can't distribute a universal quantifier over disjunction, it can only distribute over injunction. So, I would assume that I can use the logical identities like the negation laws and De Morgan's Laws, is that correct? If someone could just drop some hints or point me in the right way, I would greatly appreciate it.

Answer 1

The baisc rules are the following.

$∀$ distribute over $∧$, i.e.

$∀x \ (Px ∧ Qx) ≡ (∀x Px ∧ ∀x Qx)$,

and $∃$ distribute over $∨$, i.e.

$∃x \ (Px ∨ Qx) ≡ (∃x Px ∨ ∃x Qx)$.

In addition, we have:

$∀x \ (Px ∨ A) ≡ (∀x Px ∨ A)$, only if $x$ is not free in $A$,

and the same restriction hold for $∃$ with $∧$.

In conclusion:

$(∀x) \ [P(x) ∧ (Q(b) ∨ R(x)] ≡ (∀x)P(x) ∧ (Q(b) ∨ (∀x)R(x))$,

because we can "safely" distribute $∀$ over $∧$ and in this case we can also distribute it over $∨$, because $x$ is not free in $Q(b)$.