Answer
The pair of statements have the same truth values for every choice of P(x), Q(x), and D.
Work Step by Step
No matter what the domain D or the predicates P(x) and Q(x) are, the given statements have the same truth value. If the statement “∀x in D, (P(x) ∧ Q(x))” is true, then P(x) ∧ Q(x) is true for every x in D, which implies that both P(x) and Q(x) are true for every x in D. But then P(x) is true for every x in D, and also Q(x) is true for every x in D. So the statement “(∀x in D, P(x)) ∧ (∀x, in D, Q(x))” is true. Conversely, if the statement “(∀x in D, P(x)) ∧ (∀x in D, Q(x))” is true, then P(x) is true for every x in D, and also Q(x) is true for every x in D. This implies that both P(x) and Q(x) are true for every x in D, and so P(x) ∧ Q(x) is true for every x in D. Hence the statement “∀x in D, (P(x) ∧ Q(x))” is true
