Can I rewrite propositional formula in this way?

Question

I have a propositional formula:

$$(\neg p \lor \neg q \lor r)$$

Can i rewrite it in this way?:

$$(\neg p \lor \neg q \lor r) = (\neg p \lor (q \land \neg r))= (\neg p \lor q) \land (\neg p \lor \neg r))$$

Answer 1

No, that's incorrect. Specifically, the first claimed equality is wrong. For example, if $p$ is true and $q$ is false then $(\neg p\vee\neg q\vee r)$ is true but $(\neg p\vee (q\wedge\neg r))$ is false.

Per the comments, it seems that your goal is to convert your original formula into a $2$-CNF formula, that is, a conjunction of disjunctions of pairs of literals. This can't be done, for a very strong reason:

Can you show that in fact your formula does not imply any formula of the form $r_0\vee r_1$ for $r_0,r_1$ literals?

Answer 2

You forgot a negation in the second expression, $(\neg p \lor (q \land \neg r))$. The correct expression should be $(\neg p \lor \neg(q \land \neg r))$ (De Morgan's Laws). Hope this helps!