Home Work – Nov. 15

Recall the “Knowledge about knowledge – II” page (slide #10) from the Nov. 8 lecture.  Give an example of each of following:

1.     An [m, n] integer pair such that S cannot know m & n given only the sum S.

2.     An [m, n] integer pair such that P cannot know m & n given only the product P.

3.     An [m, n] integer pair such that S knows that P does not know m & n.

4.     An [m, n] integer pair such that P can deduce m & n from learning that S realizes that he cannot know m & n only from P.

Use semantic tableaux to prove these propositions:

1. (p v p) →p

2. q → (p v q)

3. (q → r) → [(p v q) → (p v r)]

4. [(p → q) ^ (p → r)] → [p → (q ^ r)]

5. [p → (q  ^ r)] → [(p → q) ^ (p → r)]

6. [(p → q) ^ (p → r)] ↔ [p → (q ^ r)]

7. [(p → r) ^ (q → r)] ↔ [(p v q) → r]

8. [(p → q) v (p → r)] ↔ [p → (q v r)]

9. [(p → r) v (q → r)] ↔ [(p ^ q) → r]

10. (p ↔ q) ↔ [(p ^ q) v (~p ^ ~q)]

11. ~(p ↔ q) ↔ (p ↔ ~q)