Show that ##\forall x(P(x) \land Q(x)) \equiv \forall xP(x) \land \forall xQ(x)##
The Attempt at a Solution
Based on my work from propositional logic, to show that two expressions are logically equivalent, then we have to show that ##\forall x(P(x) \land Q(x)) \Longleftrightarrow \forall xP(x) \land \forall xQ(x)## is a tautology; that is, it is always true. It is always true if they have the same truth values for all x in the domain. For propositional logic, it was a matter of listing out the finite combinations of truth values and showing that they are always the same. However, with predicate logic, we are dealing with an infinite domain of discourse, so we can't just list them off. How should I proceed then?