Predicates and Quantifiers about Show Precedence of Quantifiers

Question

Predicates and Quantifiers Questions

for all x P(x) /\ for all x Q(x)

for all x(P(x) /\ Q(x))

why is not logically equivalent?

Answer 1

The first statement:

for all x : P(x) and for all x : Q(x) 

may not refer to the same domain for x so it could be written:

1. for all x in Y : P(x) and for all x in X : Q(x) where X != Y

or

2. for all x in Y : P(x) and for all x in X : Q(x) where X = Y

whereas:

for all x : P(x) and Q(x) 

refers to the same x and is equivalent to 2 above.