CSE 20 – Discrete Mathematics

1. Peer Instruction in Discrete Mathematics by Cynthia Leeis licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.Based on a work at http://peerinstruction4cs.org.Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org. CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett

2. Today’s Topics: • Knights and Knaves • Review of Proof by Contradiction

3. 1. Knights and Knaves

4. Knights and Knaves • Knights and Knaves scenarios are somewhat fanciful ways of formulating logic problems • Knight: everything a knight says is true • Knave: everything a knave says is false

5. You approach two people, you know the one on the left is a knave, but you don’t know whether the one on the right is a knave or a knight. • Left: “Everything she says is true.” • Right: “Everything I say is true.” • What is she (the one on the right)? • Knight • Knave • Could be either/not enough information • Cannot be either/situation is contradictory

6. You approach one person, but you don’t know whether he is a knave or a knight. • Mystery person: “Everything I say is true.” • What is he? • Knight • Knave • Could be either/not enough information • Cannot be either/situation is contradictory

7. You approach one person, but you don’t know whether she is a knave or a knight. • Mystery person: “Everything I say is false.” • What is she? • Knight • Knave • Could be either/not enough information • Cannot be either/situation is contradictory

8. You meet 3 people:A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything] • This is a really tricky one, but take a moment to see if you can determine which of the following is a possible solution: • A: Knave, B: Knave, C: Knave • A: Knight, B: Knight, C: Knight • A: Knight, B: Knight, C: Knave (Suggestion: eliminate wrong choices rather than trying to solve the puzzle directly. In your groups: please discuss logic for eliminating choices.)

9. 2. Proof by Contradiction

10. Proof by Contradiction Steps • What are they? • 1. Assume what you are proving, 2. plug in definitions, 3. do some work, 4. show the opposite of what you are proving (a contradiction). • 1. Assume the opposite of what you are proving, 2. plug in definitions, 3. do some work, 4. show the opposite of your assumption (a contradiction). • 1. Assume the opposite of what you are proving, 2. plug in definitions, 3. do some work, 4. show the opposite of some fact you already showed (a contradiction). • Other/none/more than one.

11. You meet 3 people:A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything] • A: Knight, B: Knight, C: Knave • Zeroing in on just one of the three parts of the solution, we will prove by contradiction that A is a knight.

12. A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything]Thm. A is a knight. Proof (by contradiction): Assume not, that is, assume A is a knave. Try it yourself first!

13. A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything]Thm. A is a knight. Proof (by contradiction): Assume not, that is, assume A is a knave. Then what A says is false. Then it is false that at least one is a knave, meaning zero are knaves. So A is not a knave, but we assumed A was a knave, a contradiction. So the assumption is false and the theorem is true. QED.

14. You meet 3 people:A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything] • A: Knight, B: Knight, C: Knave • Zeroing in on the second of the three parts of the solution, we will prove by contradiction that B is a knight.

15. A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything]Thm. B is a knight. Proof (by contradiction): Assume not, that is, assume B is a knave. Try it yourself first!

16. A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything]Thm. B is a knight. Proof (by contradiction): Assume not, that is, assume B is a knave. Then what B says is false, so it is false that at most two are knaves. So it must be that all three are knaves. Then A is a knave. So what A says is false, and so there are zero knaves. So B must be a knight, but we assumed B was a knave, a contradiction. So the assumption is false and the theorem is true. QED.

17. A: “At least one of us is a knave.”B: “At most two of us are knaves.”[C doesn't say anything]Thm. B is a knight. Proof (by contradiction): Assume not, that is, assume B is a knave. Then what B says is false, so it is false that at most two are knaves. So it must be that all three are knaves. Then A is a knave. So what A says is false, and so there are zero knaves. But all three are knavesand zero are knaves is a contradiction. So B must be a knight, but we assumed B was a knave, a contradiction. So the assumption is false and the theorem is true. QED. We didn’t need this step because we had already reached a contradiction.