# Convert form from CNF to DNF

I have a few question about converting forms to DNF, CNF and from CNF to DNF.

1) How can I convert this to DNF $(p \vee q) \wedge (q \vee \neg r)$

2) How can I convert this to CNF $(p \wedge q) \vee (q \wedge\neg r)$

3) Is there any fast way to convert one form from CNF to DNF or DNF to CNF?

• In second item you supposed $\wedge$ in a second parentheses? – Michael Galuza Jul 1 '15 at 17:52
• yes, my fault, i edited – hadson172 Jul 1 '15 at 18:02

$$1.\quad (p\vee q)\wedge (q\vee\neg r) = q\wedge(p\vee q) \vee\neg r\wedge(p\vee q) = q \vee \neg r p\vee \neg rq = q\vee p\neg r$$ $$2. \quad (p\wedge q)\vee (q\wedge\neg r) = q(p\vee\neg r)$$

• Could you explain which rules you used to make form like this? – hadson172 Jul 1 '15 at 18:03
• @hadson172, I used: $a\wedge(b\vee c) = a\wedge b\vee a\wedge c$ (and vise versa), $a\wedge a = a$, $a\wedge (a\vee b)=a$. – Michael Galuza Jul 1 '15 at 18:16

This is mechanized in Maple. For example,

with(Logic):
Export(Normalize(&and(&or(p, q), &or(q, &not(r))))), form = DNF));


$$p \land q \lor p \land \neg r \lor q \lor q \land \neg r$$ See ?Logic for info.

PS. It should be noted that the original Maple input is not exactly represented in the above code.