1
$\begingroup$

I am trying to convert the following to CNF:

$$(P → (Q → R)) → (P → (R → Q))$$

I've done some of it but I'm unsure of what I am supposed to do after this. Here is what I have gotten so far.

$$¬(P \to (Q \to R)) ∨ ( P \to (R \to Q)$$

$$¬(P \to (¬Q ∨ R)) ∨ ( P \to (¬R ∨ Q) $$

$$¬(¬P ∨ (¬Q ∨ R)) ∨ ( ¬P ∨ (¬R ∨ Q) $$

$$(P ∧ ¬(¬Q ∨ R)) ∨ ( ¬P ∨ (¬R ∨ Q) $$

$$(P ∧ (Q ∧ ¬R)) ∨ ( ¬P ∨ (¬R ∨ Q) $$

$$(P ∧ Q ∧ ¬R) ∨ ( ¬P ∨ ¬R ∨ Q) $$

$$(P ∧ Q ∧ ¬R) ∨ ¬P ∨ ¬R ∨ Q $$

$$\neg P \lor \neg R \lor Q$$

I just don't know what to do when I get here. Any help will be much appreciated. Thanks!

  • 0
    $\neg P\vee\neg R\vee Q$ *is* conjunctive normal: it is a conjunctive sequence of terms that are purely disjunctions of literals (or negations of such). There is just one such term in the sequence, is all. – 2017-01-31

1 Answers 1

1

You can drop the second set of parentheses, and the $P \land Q \land \neg R$ term gets absorbed by $\neg R$, which leaves you with:

$\neg P \lor \neg R \lor Q$

which is in CNF!

So use the following general equivalence principle:

Absorption

$P \lor (P \land Q) \Leftrightarrow P$

  • 0
    So, ¬R is logically equivalent to P∧Q∧¬R, meaning we can drop that and be left with ¬P∨¬R∨Q? How do we know that the ¬R is the one that is logically equivalent and not ¬P or Q? – 2017-01-31
  • 0
    @name. not quite: $\neg R$ is equivalent to $(P \land Q \land \neg R) \lor \neg R$. And yes, we could also have absorbed that term using $Q$ ... But not with the $\neg P$ – 2017-01-31
  • 0
    I think I understand it but I am not sure where the ¬P came from. NVM, just saw your edit. – 2017-01-31
  • 0
    @name. That one was the one you got from converting the $P \rightarrow (Q \rightarrow R)$ ... Which you all did correctly by the way! – 2017-01-31
  • 0
    Thank your for all your help. I believe I understand it now, but I am not sure how exactly to write that in my solution. May you check my edited post and give me feedback on how I wrote the solution? Thanks :) – 2017-01-31
  • 0
    Not quite right ... I'll edit it! – 2017-01-31
  • 0
    Thank you so much for all your help!!! – 2017-01-31
  • 0
    @name no problem ... Do you see what happens at the end and how that is an example of Absorption? – 2017-01-31