Conjuctive Normal Form

Question

In Boolean logic, a formula is in conjunctive normal form or clausal normal form if it is a conjunction of clauses, where a clause is a disjunction of literals; otherwise put, it is an AND of ORs.

I See that this formula

in CNF form is equivalent to

I try to convert it, but I'm failed. I think maybe it's not equivalent. one expert please help me step by step.

Answer 1

First we have to "unwid" $(¬A→B)↔¬C$ as :

$[(¬A→B)→¬C)]∧[¬C→(¬A→B)]$

and then use the equivalence between :

$(¬A→B)$ and $(A∨B)$

and :

$(A→B)$ and $(¬A∨B)$.

Thus :

(i) $(¬A→B)↔¬C$

(ii) $[(¬A→B)→¬C)]∧[¬C→(¬A→B)]$

(iii) $[(A \lor B) \rightarrow \lnot C] \land [\lnot C \rightarrow (A \lor B)]$

(iv) $[\lnot (A \lor B) \lor \lnot C] \land [C \lor (A \lor B)]$

(v) $[(\lnot A \land \lnot B) \lor \lnot C] \land (A \lor B \lor C)$ --- by De Morgan's laws

(vi) $[(\lnot A \lor \lnot C) \land (\lnot B \lor \lnot C)] \land (A \lor B \lor C)$ --- by Distributive laws