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
(b ≡ c) mean ( b כ c) ^ (c כ b)
But if a,b,c are sets, then (b ≡ c) is either T or F. You cannot then say
a כ T
So, I guess a,b,c are all T/F values, and using subset notation does not seem correct.
Maybe I misunderstand your ≡ operator.
you did not misunderstand the ≡ operator
my problem has nothing to do with assigning truth values though I have to rewrite it in conductive normal form
ah. Then does b כ c mean implies? If so, then your initial expression appears correct.
Sorry, but I don't know the conductive normal form. Surely google can help with this.
To convert the formula (a ≡ (b ≡ c)) to conjunctive normal form (clause form), we need to follow a series of steps:
Step 1: Eliminate the bi-implication symbol (≡) using the following equivalence:
(a ≡ b) ≡ c is equivalent to (a → b) ∧ (b → a) ∧ (b → c) ∧ (c → b)
Step 2: Distribute ∧ over ∨:
(a → b) ∧ (b → a) ∧ (b → c) ∧ (c → b) is equivalent to
(¬a ∨ b) ∧ (¬b ∨ a) ∧ (¬b ∨ c) ∧ (¬c ∨ b)
Step 3: Collect terms using parentheses:
(¬a ∨ b) ∧ (¬b ∨ a) ∧ (¬b ∨ c) ∧ (¬c ∨ b) can be organized as
(a ∨ ¬b) ∧ (a ∨ b) ∧ (b ∨ ¬c) ∧ (b ∨ c)
Step 4: Convert to clause form:
(a ∨ ¬b) ∧ (a ∨ b) ∧ (b ∨ ¬c) ∧ (b ∨ c) can be simplified to
(a ∨ ¬b) ∧ (b ∨ ¬c) ∧ (a ∨ b) ∧ (b ∨ c)
Thus, the conjunctive normal form (clause form) representation of the formula (a ≡ (b ≡ c)) is:
(a ∨ ¬b) ∧ (b ∨ ¬c) ∧ (a ∨ b) ∧ (b ∨ c)