Question

Suppose that some knowledge base contains various propositional-logic sentences that utilize symbols A, B, C, D (connected with various connectives). There are only two cases when the knowledge base is false:
- First case: when A is true, B is false, C is false, D is true.
- Second case: when A is false, B is true, C is true, D is false.

In all other cases, the knowledge base is true. Write a conjunctive normal form (CNF) for the knowledge base.

Answer 1

Sol)

case 1 : A is true, B is false, C is false, D is true

case 2: A is false, B is true, C is true, D is false

Knowledge base is false in above two cases, true in all other remaining cases

so, the truth table for above scenario

A	B	C	D	X
T F	F T	F T	T F	F F
For all remaining	T

To write CNF(Conjunctive Normal Form), we take rows that contain X= F( false rows), then we negate each row and return conjuction over these negated rows.

$\sim(A \wedge \sim B \wedge \sim C \wedge D ) \wedge \sim (\sim A \wedge B\wedge C\wedge \sim D)$

We apply demorgan's law and law of double negation to produce disjunction

$(\sim A \vee B \vee C \vee \sim D ) \wedge (A \vee \sim B\vee \sim C \vee D)$

so, the final CNF is

$(\sim A \vee B \vee C \vee \sim D ) \wedge (A \vee \sim B\vee \sim C \vee D)$

In other way we can write it as

'.'(dot symbol represents the conjunction $\wedge$ )

+(symbol represents the disjunction $\vee$ )

__(bar symbol represents the negation)