Question

Which of the following predicate calculus statements is/are valid

• A. $\left(\forall x\right)P\left(x\right)\vee \left(\forall x\right)Q\left(x\right)\to \left(\forall x\right)\{P\left(x\right)\vee Q\left(x\right)\}$
• B. $\left(\exists x\right)P\left(x\right)\wedge \left(\exists x\right)Q\left(x\right)\to \left(\exists x\right)\{P\left(x\right)\wedge Q\left(x\right)\}$
• C. $\left(\exists x\right)\{P\left(x\right)\vee Q\left(x\right)\}\to \left(\forall x\right)P\left(x\right)\vee \left(\forall x\right)Q\left(x\right)$
• D. $\left(\exists x\right)\{P\left(x\right)\vee Q\left(x\right)\}\to \left(\exists x\right)P\left(x\right)\vee \forall xQ\left(x\right)$

Answer 1

The correct option is A $\left(\forall x\right)P\left(x\right)\vee \left(\forall x\right)Q\left(x\right)\to \left(\forall x\right)\{P\left(x\right)\vee Q\left(x\right)\}$

According to distributive properties
$\forall x(P\left(x\right)\wedge Q\left(x\right))\leftrightarrow \forall xP\left(x\right)\wedge \forall xQ\left(x\right)$

$\forall x(P\left(x\right)\vee \forall Q\left(x\right)\to \forall x(P\left(x\right)\vee Q\left(x\right))$

$\exists x(P\left(x\right)\vee Q\left(x\right))\leftrightarrow \exists xP\left(x\right)\vee \exists xQ\left(x\right)$

$\exists x(P\left(x\right)\wedge Q\left(x\right))\to \exists xP\left(x\right)\wedge \exists xQ\left(x\right)$
So option (a) is valid.