Give a proof or counterexample, whichever is appropriate.
1. NOT (?x, (P(x) OR Q(x) OR R(x))) is logically equivalent to?x, ((NOT P(x)) AND (NOT Q(x)) AND (NOT R(x))).
2. NOT (?x, (P(x) AND Q(x) AND R(x))) is logically equivalent to?x, ((NOT P(x)) OR (NOT Q(x)) OR (NOT R(x))).
3. NOT (?x, (P(x) ? Q(x))) is logically equivalent to ?x, (P(x)?NOT Q(x)).
4. NOT (?x, (P(x) ? Q(x))) is logically equivalent to ?x, (P(x)AND (NOT Q(x))).
5. ?x, (P(x) OR Q(x)) is logically equivalent to (?x, P(x)) OR(?x, Q(x)).
6. ?x, (P(x) AND Q(x)) is logically equivalent to (?x, P(x)) AND(?x, Q(x)).
7. ?x, (P(x) OR Q(x)) is logically equivalent to (?x, P(x)) OR(?x, Q(x)).
8. ?x, (P(x) AND Q(x)) is logically equivalent to (?x, P(x)) AND(?x, Q(x)).
9. ?x, (P (x) ? Q(x)) is logically equivalent to (?x, (x)) ?(?x, Q(x)).
