Description
Logic
Question 1.
Simplify the following statements (so that negation only appears right before variables).
Question 2.
Consider a vocabulary with only four propositions, A, B, C, and D. How many models are there for the following sentences? How did you derive your result?
- ¬A∨B∨¬C
- (A ⇒ B)∧¬B∧C.
Question 3.
Convert the following to CNF with procedures:
((~S => P) ^ (R=>Q)) => (~P => Q)
Question 4:
Decide whether each of the following sentences is valid, unsatisfiable, or neither. Verify
your decisions using truth tables
For each of the following questions: 5.
- Convert the information given into logic statements.
- Prove that the proposition(s) are valid using a truth table (Excel may help with this).
- Referencing your proofs, explain whether or not the proposition(s) is/are valid.
Question 5.
A bank was robbed and Inspector Craig and Sergeant McPherson were on the case trying to establish the guilt or innocence of four suspects Alice, Bob, Carol and Dave. The nefarious characters are the only people who could be involved in these bank robberies and at least one of them is guilty. In each case the Inspector and Sergeant establish certain facts.
Write an argument in words to establish the guilt or innocence of Alice, Bob and Carol and Dave. Note that the clues provided may not be sufficient to determine the guilt and innocence of all of the suspects, but should be sufficient to establish the guilt of at least one person. Say that we establish that:
(1) If Alice was guilty, then she had exactly one other accomplice.
(2) Bob and Carol were both together at the time of the crime.
(3) If exactly two are guilty then Alice is one of them.
(4) Bob and Dave never work together.
(5) If both Bob and Carol were not involved then Dave is guilty.
Translate each of the clues to a truth valued sentence using the connectives and, or, not and if . . . then and the propositions: A representing the statement “Alice is guilty,” B representing the statement “Bob is guilty,” C representing “Carol is guilty,” and D representing “Dave is guilty.” Create a truth table establishing the truth values of the clues in terms of the truth values of A, B, C and D
For each of the following questions: 6-7.
- Convert the information given into logic statements.
- Convert the logic statements into conjunctive normal form.
- Prove that the proposition(s) are valid using a truth table (Excel may help with this).
- Provide a second proof using proof by resolution using refutation.
- Referencing your proofs, explain whether or not the proposition(s) is/are valid.
Question 6:
Detective James is solving a case. The four suspects A, B, C and D made the following statements:
- If A is guilty then B is guilty
- If B is guilty then C is guilty or A is innocent
- If D is guilty then A is guilty and C is innocent
- If D is guilty then A is guilty
Is D guilty based on these statements?
Question 7:
If tomorrow is a snow storm and the zipper is mended, Patrick will wear a heavy coat. If the zipper is not mended, Patrick will not wear heavy coat. It will be a snow storm tomorrow and the zipper is not mended. Will Patrick wear a heavy coat?
