Show that AxPx v AxQx and AxPx v Qx is not logically equival

Show that AxP(x)) v AxQ(x) and Ax(P(x) v Q(x)) is not logically equivalent.

Solution

Consider a domain of discourse D with two elements, a and b, such that P(a) is true and Q(a) is false, while P(b) is false and Q(b) is true. Then the propositions x P(x) is false, since P(b) is. Similarly, since Q(a) is false then so is x Q(x). Because both propositions are false, the biconditional x P(x) x Q(x) is true. However, the biconditional P(a) Q(a) is false since the two prositions have different truth values. (The same can be said for P(b) Q(b)).) Consequently, the statement x (P(x) Q(x)) is false. Since x (P(x) Q(x)) is false for this domain D while x P(x) x Q(x) is true, the two statements are not logically equivalent.