Convert the following formula to conjunctive normal form (clause form). a)
a ≡ (b ≡ c)
is the answer
(a כ(( b כ c) ^ (c כ b))) ^ ((( b כ c) ^ (c כ b)) כ a)
unsure thanks
To convert a formula to conjunctive normal form (CNF), we need to apply several logical equivalences and transformations.
Starting with the formula a ≡ (b ≡ c), we can break it down into two separate implications to simplify it further:
a ⇒ (b ≡ c) ... (1)
(b ≡ c) ⇒ a ... (2)
Let's focus on converting each implication into CNF:
1. Converting a ⇒ (b ≡ c) to CNF:
We'll use the following equivalence: p ⇒ q ≡ ¬p ∨ q
¬a ∨ (b ≡ c) ... (3)
2. Converting (b ≡ c) ⇒ a to CNF:
Similarly, using the equivalence: p ⇒ q ≡ ¬p ∨ q
¬(b ≡ c) ∨ a ... (4)
Next, we'll work on transforming (3) and (4) into CNF by distributing the disjunction (∨) over conjunction (^).
3. Distributing ∨ over ^ in ¬a ∨ (b ≡ c):
(¬a ∨ b) ≡ (¬a ∨ c) ... (5)
4. Distributing ∨ over ^ in ¬(b ≡ c) ∨ a:
(¬b ∨ ¬c ∨ a) ... (6)
Finally, we combine the CNF forms (5) and (6) by taking the conjunction (^) of both results:
(¬a ∨ b) ≡ (¬a ∨ c) ^ (¬b ∨ ¬c ∨ a)
This is the converted formula in conjunctive normal form (CNF).
So, the correct answer is:
(a ≡ ((b ≡ c) ^ (c ≡ b))) ^ (((b ≡ c) ^ (c ≡ b)) ≡ a)