SE 500 Mathematics for Software Engineering
Fall 2009
HW #7: Applications of Propositional Calculus
Due: November 2

All exercises cited by number come from the Gries & Schneider text.

1. Use resolution (in the proof by contradiction mode, as illustrated in class and on the relevant web page) to show that the following argument is valid. (This is adapted from Exercise 5.1(d).)

Joe hates either Mary or Larry. If Joe loves Mary or hates Larry, either Mom or Dad is sad. If Mom is sad, so is Dad. Dad is happy. Therefore, Joe hates Mary.

Associate identifiers with the primitive propositions as follows: 

         HM : Joe hates Mary
         HL : Joe hates Larry
         MS : Mom is sad
         DS : Dad is sad

For the purposes of this problem, assume that love and hate are mutually exclusive and are the only two feelings that one person can have toward another. Thus, rather than introducing another identifier for the proposition "Joe loves Mary", take this proposition to be equivalent to ¬HM. Similarly, take the proposition "Dad is happy" to be equivalent to ¬DS.

2. Prove the argument given in the problem above using the methods from the Gries & Schneider text. Keep in mind that any premise (and the conclusion, for that matter) can be rewritten as any boolean expression to which it is equivalent. (In particular, you mind find it helpful to make use of the contrapositive of an implication.)

The remaining problems are with respect to the Island of Knights and Knaves. Heed Gries's advice: If (inhabitant) B utters a proposition that is formalized by the expression X, this gives rise to the premise b ≡ X (because X is true if and only if b is true (i.e., B is a knight)).


3. Do Exercise 5.6(c): You ask B whether there is gold on the island. He replies, "There is gold on the island if and only if I am a knight." Can it be determined whether B is a knight? Can it be determined whether there is gold on the island?

4. Do Exercise 5.6(e): B, C, and D are sitting together. C says, "There is (exactly) one knight among us." D retorts, "You're lying." What can you determine about the knighthood or knavehood of the three?

Hint: The expression

(b ≡ c ≡ d) ∧ ¬(b ∧ c ∧ d)

corresponds to C's statement. (The first conjunct says that either one or three of them are knights and the second conjunct rules out all three being knights.)

5. Do Exercise 5.6(f): A stranger asks a question of B, but B's reply is too indistint to be heard. So the stranger asks C, "What did he say?", to which C replies, "B said that there is (exactly) one knight among us." Then D chimed in with "Don't believe C; he's lying!"

What are C and D?