Question

Show that ∀xP(x)∨∀xQ(x) and ∀x∀y(P(x)∨Q(y)), where all quantifiers have the same nonempty domain, are logically equivalent.

Answer 1

Use algorithm of reduction $\forall xP(x)\vee\forall xQ(x)$ to a prenex normal form.

1)$\forall xQ(x)$ is equivalent to$\forall yQ(y)$, so $\forall xP(x)\vee\forall xQ(x)$ is equivalent to $\forall xP(x)\vee\forall yQ(y)$

So since $\forall yQ(y)$ does not contain the variable $x$, we have $\forall xP(x)\vee\forall yQ(y)$ is equivalent to$\forall x(P(x)\vee\forall yQ(y))$

So $\forall xP(x)\vee\forall xQ(x)$ is equivalent to $\forall x(P(x)\vee\forall yQ(y))$

2)Consider $P(x)\vee\forall yQ(y)$.

It is equivalent to $\forall yQ(y)\vee P(x)$.

Since $P(x)$ does not contain the variable $y$, we have $\forall yQ(y)\vee P(x)$ is equivalent to $\forall y(Q(y)\vee P(x))$.

And since $Q(y)\vee P(x)$ is equivalent to $P(x)\vee Q(y)$, we have $\forall y(Q(y)\vee P(x))$ is equivalent to $\forall y(P(x)\vee Q(y))$

So $P(x)\vee\forall yQ(y)$ is equivalent to $\forall y(P(x)\vee Q(y))$

3)From step 2 we have that $\forall x(P(x)\vee\forall yQ(y))$ is equivalent to $\forall x\forall y(P(x)\vee Q(y))$

4)So step 1 and step 3 gives us equivalence of $\forall xP(x)\vee\forall xQ(x)$ and $\forall x\forall y(P(x)\vee Q(y))$