3. (16 marks Here is a list of five propositional forms and three circuits. This list contains four pairs of logically equivalent entries (we will say a circuit is equivalent to a propositional form if the propositional form describes the value of the circuit's output for all possible combinations of input values). For instance, maybe A = B, C = D, E = F and G = H). First determine the four pairs, and then prove for each pair that one element of the pair is logically equivalent to the other one. (A) P+(q+r). (B) 9+(p+r). (C) (p^q) +(q Vr). (D) ~(~p V r) +(r^q). (E) (p+r)^(~~(qV~p)) (F) PO output qop output (H) -06 output You are allowed to use truth tables to figure out the equivalences; however at least three of your proofs must use a sequence of known logical equivalences (see Epp 5 or Epp-4 table 2.1.1, Epp-3 table 1.1.1, Rosen-6 table 6 in section 1.2, Rosen-7 table 6 in section 1.3, or Dave's excellent formula sheet; you can also assume that x + y = ~3 Vy and that x y = (x Vy) A ~(2 Ay) = (~2 Ay) v (x^~y). The fourth proof can use either a sequence of known logical equivalences, or a truth table. Hint: you might want to translate the circuits into formulas that mirror exactly as the circuit is implemented, before using any logical equivalence rules