A formula is said to be in conjunctive normal form (CNF) if it is a conjune tion of Clauses. A Clause is a disjunction of atoms or their negations. Infor- mally, a formula is in conjunctive normal form if it can be written as clauses connected with the and operator, and a formula is a clause if the atoms and their negations are connected using or operations. For example: .av-b, a, 1, and a Vbveva are all examples of clauses whereasa, -(a vb), a +b are not examples of clauses. Any clause or conjunction of clauses are said to be in conjunctive normal form. So a V-6, (a v-b) (a Vb), a bien are all in conjunctve normal form but the formula a Ab Ve is not. This last formula is not in conjunctive normal form as A takes precedence over V, hence we see (a b) Ve is not a conjunction of clauses. Applying distributivity to (Ab) ve we get the logically equivalent formula, (a vb) A (a V e) which is in conjunctive normal form (it is the conjunction of two clauses) The only operators that can be used in a formula in conjunctive normal form are and, or, and not. Every formula has a logically equivalent formula in conjunctive normal form. Conjunctive normal form finds many applications in computer science in cluding areas such as automated theorem proving and circuit theory. Moreover, the famous k-sat problem involves finding a configuration of truth values which satisfies a formula in conjunctive normal form, whose clauses contain at most k variables. For each of the below formulas, write a logically equivalent formula in con- junctive normal form. Simplify each clause in your final answer. For example the formula (a V-V-6) (e v dvd) should be simplified to (av-), since (a V-V-6) = (a V-5) by the absorbtion law, evdvd = CVT = T by excluded middle and domination laws, and finally (a V-b) AT =av-6, by the identity law. a) (d) b)-(-Vove) c) (a1-6) (end) d) V (Ac) V-(eva)