Problem 3. (15 points) Using the laws of logic and the rules of inference for quantified propositions, prove that the pairs of quantified propositions below are equivalent. You may assume that all variables and predicates are defined over a common non-empty domain. First prove that AB and then also prove that BA. a. (5 points) A:x(P(x)Q(x)) and B:xP(x)yQ(y) To get started, here is the proof that AB : 1. x(P(x)Q(x)) 2. P(a)Q(a) Universal Instantiation, 1 3. P(a) Simplification, 2 4. xP(x) Universal Generalization, 3 (since a is an arbitrary element of the domain). 5. Q(a) Simplification, 2 6. yQ(y) Universal Generalization, 3 7. xP(x)yQ(y) Conjunction, 4,6 b. (10 points) A: x(P(x)Q(x)) and B: xP(x)yQ(y)