Using Conjunctive Normal form to find when wff is true

  • Thread starter ChiralSuperfields
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In summary, setting ##P_1## and ##P_2## to true will make the CNF true because the negation of ##P_2## will become true and the other clauses will also evaluate to true.
  • #1
ChiralSuperfields
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Homework Statement
I am trying to understand how to tell which values for a proposition need to be taken on to make a CNF true.
Relevant Equations
CNF notation
For this,
1689831998459.png

Does someone please know how setting ##P_1## and ##P_2## true makes the CNF true? If I see ##P_2## true, then it ##(true + false)## since it is negated. Therefore, should they be setting ##P_1## true and ##P_2## false?

Many thanks!
 
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  • #2
ChiralSuperfields said:
Homework Statement: I am trying to understand how to tell which values for a proposition need to be taken on to make a CNF true.
Relevant Equations: CNF notation

For this,
View attachment 329444
Does someone please know how setting ##P_1## and ##P_2## true makes the CNF true? If I see ##P_2## true, then it ##(true + false)## since it is negated. Therefore, should they be setting ##P_1## true and ##P_2## false?

Many thanks!
As it says, "+" specifies OR (##\vee##),
so (true + false) = (true OR false) = true.
So setting P1 and P2 true gives:
(T + F) (T + ? + ?) (T + ?)
which is (T) (T) (T)
since (True OR anything) is True
so the overall result is T.
 
  • Informative
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1. What is Conjunctive Normal Form (CNF)?

Conjunctive Normal Form is a way of representing logical statements in propositional logic. It is a conjunction of clauses, where each clause is a disjunction of literals (either a variable or its negation).

2. How do you convert a logical statement to CNF?

To convert a logical statement to CNF, you need to use the following steps:

  1. Eliminate any implications and bi-implications using logical equivalences.
  2. Move negations inwards using De Morgan's laws.
  3. Use distributive laws to eliminate any disjunctions over conjunctions.
  4. Apply double negation elimination to remove any double negations.
  5. The resulting statement will be in CNF.

3. How do you determine if a statement is true in CNF?

To determine if a statement is true in CNF, you need to check if all of its clauses are true. If a clause contains a literal and its negation, then the clause is false. If all clauses are true, then the statement is true.

4. Can any logical statement be converted to CNF?

Yes, any logical statement can be converted to CNF using the steps mentioned above. However, the resulting statement may contain a large number of clauses and may not be easily readable.

5. Are there any benefits to using CNF to determine the truth of a statement?

Yes, using CNF can make it easier to determine the truth of a statement as it reduces the number of possible combinations to check. Additionally, it can also help identify any contradictions or inconsistencies in the statement. CNF is also a standard form used in automated theorem proving and logic programming.

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