# Convert this FORMULA to CNF and DNF

I'm having serious problems whenever I try to convert a formula to CNF/DNF. I can do it with easier formulas, but this one I just can't understand for some reason:

(A∧¬B→B∧¬C)→¬A∨B


I've tried removing the → operators and replacing them with other formulas, after that I've tried to apply De Morgan's laws to it, etc, like this:

1. ¬A∨B∨¬(A∧¬B→B∧¬C)
2. ¬A∨B∨¬((B∧¬C)∨¬(A∧¬B))
3. ¬A∨B∨(¬(B∧¬C)∧¬¬(A∧¬B))
4. ¬A∨B∨((¬B∨¬¬C)∧(A∧¬B))
5. ¬A∨B∨((¬B∨C)∧(A∧¬B))


I'm not sure if I even did one correctly, not the mention all of it, and I'm also stuck here, so I don't know how to continue..

I'd really appreciate if someone could help me solve this! Thanks!

So far so good!

Taking it from where you left:

$$\neg A \lor B \lor ((\neg B \lor C) \land (A \land \neg B)) \overset{Association}{=}$$

$$(\neg A \lor B) \lor ((\neg B \lor C) \land A \land \neg B) \overset{Distribution}{=}$$

$$[(\neg A \lor B) \lor (\neg B \lor C)] \land [(\neg A \lor B) \lor A)] \land [(\neg A \lor B) \lor \neg B] \overset{Association}{=}$$

$$[\neg A \lor B \lor \neg B \lor C] \land [\neg A \lor B \lor A] \land [\neg A \lor B \lor \neg B] \overset{Association, Commutation}{=}$$

$$[\neg A \lor C \lor (B \lor \neg B)] \land [B \lor (A \lor \neg A)] \land [\neg A \lor (B \lor \neg B)] \overset{Complement}{=}$$

$$[\neg A \lor C \lor \top] \land [B \lor \top] \land [\neg A \lor \top] \overset{Annihilation}{=}$$

$$\top \land \top \land \top \overset{Idempotence}{=}$$

$$\top$$

... which is both in DNF and CNF

• Thank you so much! I guess I can use double "((", instead of "[" right? My teacher is pretty oldschool and it's not allowed for us, to use [ ] brackets for some reason, etc – Arany Sandor Oct 8 '19 at 21:41
• @AranySandor You're welcome! And yes, then just use extra ( and )'s – Bram28 Oct 8 '19 at 21:43