Incorrect Answer.
The semantic entailment
x (P(x) 
Q(x)) 
x P(x) x Q(x)
is valid in predicate logic if, and only if all models M (that have
relations PM and QM defined) that satisfy
x (P(x) Q(x))
also satisfy
x P(x) x Q(x).
- A model M satisfies x (P(x) Q(x))
if, and only if the intersection of the sets PM and QM
does not equal the set of values, A, of the model.
- A model M satisfies x P(x) x Q(x)
if, and only if neither the set PM, nor the set QM equals
the set A of values of M.
With these insights at hand, it is not hard to come up with a
counterexample. Essentially, we only need to ensure
that either PM or QM equals A, and that
their intersection is different from A.
So let the model M be given
by
A = {a,b};
PM = {a,b}
QM =
{b}.
Please verify that we have
M x (P(x) Q(x)), but that we do not have
M
x P(x) x Q(x). Thus, the semantic entailment
above is not valid in predicate logic.
Back to Question.
Next Question.