# Is this the disjunctive and conjunctive nromal form for my porpositional formula F?

I want to bring the following formula $F = A \land (A \lor B) \Rightarrow (B \lor C)$ into conjunctive normal form (CNF) and disjunctive normal form. Therefore, I applied the following 5 step transformation process:

1. Step: Removing the implication:

$$\Leftrightarrow \neg (A \land(A \lor B)) \lor (B \lor C)$$

1. Step: Resolve the negation: $$\Leftrightarrow \neg A \lor (\neg A \land \neg B) \lor (B \lor C)$$

2. Step: Remove parenthesis to receive DNF $$\Leftrightarrow \neg A \lor (\neg A \land \neg B) \lor B \lor C$$

3. Step: Resolve the absorption based on step 2: $$\neg A \lor (B \lor C)$$

4. Step Removing parenthesis to have DNF as well as CNF: $$\neg A \lor B \lor C$$

I just have three questions to my normal forms:

1. Is this transformation process correct?
2. Is the formula of step 3: $\neg A \lor (\neg A \land \neg B) \lor B \lor C$ in DNF?
3. Are DNF and CNF of step 5 correct? Is it necessary to remove the parenthesis for DNF and CNF?
• In step 2, the $\lor$s should become $\land$s. – Joshua Tilley Jul 23 '18 at 17:20
• @JoshuaTilley thanks for your hint! I corrected it. Could you also please answer my questions? Thanks in advance! – user3352632 Jul 24 '18 at 7:01
• I'm not sure I know what you're asking, sorry. Also, you need to carry the change into steps 4 and 5, they should be $\sim A \lor B \lor C$ – Joshua Tilley Jul 24 '18 at 12:50
• @JoshuaTilley thanks, I updated the equations. So the first question is answered by you. thanks a lot. I hope here is anybody who can answer the 2. one! – user3352632 Jul 24 '18 at 13:21
• The answer to 2 is yes, since it is a disjunction of conjunctions. As for 3, the parentheses make no difference in this case, since $\lor$ is associative. – Joshua Tilley Jul 24 '18 at 13:26