Unit 10 Answers

1. To show these are not logically equivalent, we are going to give an example in which they take different values. Let P(x) be the statement " x is positive ", and let Q(x) be the statement" x is negative" with universe of discourse the set of integers.  Then xP(x) x Q(x) is true, but x (P(x) Q(x) ) is false.

2. Let y be a variable that does not appear in P(x).   Then x P(x) x Q(x)  is equivalent to x P(x) y Q(y).   Since y is not in P(x),   y   does not affect P(x).   Hence   y   can be brought in front of P(x).   Hence x P(x) y Q(y)   is equivalent to x y ( P(x) Q(y) ).