Explain how to read off a CNF for propositional formula directly from a DNF from its complement.
I've managed to explain it in words, but can't write a rigorous proof of that. How to show this in mathematical notations? Thanks.
Explain how to read off a CNF for propositional formula directly from a DNF from its complement.
I've managed to explain it in words, but can't write a rigorous proof of that. How to show this in mathematical notations? Thanks.
As Fizz pointed out in comments, we can use De Morgan's laws. I'll post it, since someone might need it.
Explanation: Let $P$ be propositional formula in question, $\neg P$ its complement, and $S$ denotes DNF of $\neg P$. $$S::=Q_1 \vee \cdots \vee Q_k, $$ where $Q_i,(1 \leq i \leq k)$ is AND-clause(s). Applying De Morgan's (distribution of $\neg$ over $\vee$) rule to S, we get $$\neg S::= Q_1' \wedge \cdots \wedge Q_k',$$ where $Q_i'::=\neg Q_i, (1\leq i \leq k). $ Since negating AND-clauses produce OR-clauses, we have AND of OR-clauses in $\neg S$, which is exactly CNF for P.
Thus we've showed how to get CNF of propositional formula from DNF of its complement: Negating DNF of complement gives CNF of proposition. df