3. [12 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 AB,CD, EF and GH). First determine the four pairs, and then prove for each pair that one element of the pair is logically equivalent to the other one. Two of the pairs will be very difficult to prove using rules of logical equivalence. The first one of the difficult pairs that you prove will be graded as a normal part of this question. The second difficult pair is optional; an attempt to prove this will be graded for up to a maximum of 2 points for bonus credit on this assignment. (A) (ab)(bc) (B) (a(bc))(a(bc)) (C) (abc)(abc) (D) (ba)(ca) (E) b(ac) (F, (C (I 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 xyxy and that xy(xy)(xy)(xy)(xy)). 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