SE 500 (Math for SE)   Fall 2023
HW #5: Implication and Knights & Knaves
Due: 4:00pm, Friday, October 6

The exercises and theorems referred to by number come from Chapter 3 of the Gries & Schneider text. In proving a numbered theorem, you may make use of only lower-numbered theorems (or metatheorems). Note that some of the exercises may have hints that appear in the book, but not here. Among the theorems that are most useful in developing proofs involving disjunction, conjunction and implication are (3.32), (3.35), (3.48), (3.57), (3.59), and (3.60). Make sure to respect the relative precedences of operators, which are listed on the textbook's inside front cover (in the hardback version at least) and on a web page to which there is a link from the course web page.

1. Do Exercise 3.42, which is to prove

(3.60) p ⇒ q  ≡  p ∧ q  ≡  p

2. Do Exercise 3.45, which is to prove

p ⇒ q  ≡  ¬p ∨ ¬q  ≡  ¬p

3. Do Exercise 3.48, which is to prove

(3.64) p ⇒ (q ⇒ r)  ≡  (p ⇒ q) ⇒ (p ⇒ r)

4. Do Exercise 3.50, which is to prove

p ∧ (p ⇒ q)  ≡  p ∧ q

The next two problems are to prove theorems that are implications. Rather than showing each one to be equivalent to an already-known theorem, it is suggested that you transform the left-hand side into the right-hand side, where the operator connecting each line to the next is either = or ⟹. (At this point, the only theorems available to you by which you can connect one line to the next by ⟹ are (3.76a)—(3.76c).)

5. Do Exercise 3.63, which is to prove Weakening/strengthening

(3.76d) p ∨ (q ∧ r)  ⟹  p ∨ q

6. Prove Modus tollens: (p ⟹ q) ∧ ¬q  ⟹  ¬p

7. Do Exercise 3.66, which is to prove

(3.78) (p ⇒ r) ∧ (q ⇒ r)  ≡  (p ∨ q ⇒ r)

The remaining problems concern Bill and Carol, who are natives of the Island of Knights and Knaves. Every native of the island is either a knight or a knave. Every proposition uttered by a knight is true, and every proposition uttered by a knave is false. Use b and c, respectively, to stand for the propositions Bill is a knight and Carol is a knight.

For each scenario described, make the strongest statement you can regarding the status (knight or knave) of each of Bill and Carol. Such a statement need not be so strong as to identify the status of each of Bill and Carol. For example, a scenario may be such that the most you can say is that Bill is a knight (while saying nothing about Carol), or that at least one of them is a knight. (A given scenario can, theoretically, correspond to any of the sixteen two-argument boolean functions.)

8. Bill said, "If Carol is a knave, then I am a knight."

9. A visitor to the island, upon encountering Bill and Carol, asked, "Which of you is a knight?". Carol replied, but so softly that the visitor could not hear the answer. Bill then blurted out, "Carol said that if she is a knight, then so am I".

10. A visitor to the island, upon encountering Bill and Carol, asked, "Which of you is a knight?". Carol replied, but so softly that the visitor could not hear the answer. Bill then blurted out, "Carol said that if she is a knave, then I am a knight".