I am struggling to convert this sentence to CNF:
(A ∨ B) ⇔ (C ∧ D).
I have already tried to use the Biconditional elimination logic rule to eliminate the ⇔.
(A ∨ B) → (C ∧ D) ∧ (C ∧ D) → (A ∨ B).
Then I eliminated the → with the Implication elimination logic rule. Now I have
¬(A ∨ B) ∨ (C ∧ D) ∧ ¬(C ∧ D) ∨ (A ∨ B).
I am pretty much stuck here. My professor says I should use Distributivity rule to reduce the sentence. I can't seem to find anything that matches the requirements of Distributivity rule. So, I can't seem to use Distributivity rule before doing some logical rule that I do not know of.
What am I missing here? Can Stack Overflow help me to resume the conversion to CNF?
You began with the expression:
You tried to perform the first few steps. Here I added brackets to be clear and correct:
Apply the De Morgan negation law to ¬(A ∨ B) and ¬(C ∧ D):
Simplify the right half:
The distributive law for ∨ over ∧ states that: X ∨ (Y ∧ Z) ⇔ (X ∨ Y) ∧ (X ∨ Z).
We apply the law to the left half, with X = (¬A ∧ ¬B), Y = C, Z = D:
Apply the distributive law to two subexpressions in the left half:
Remove the extra brackets because ∧ is associative and commutative:
Rearrange the variables, and we have our final formula in conjunctive normal form (CNF):