LOGIC. ON THE ISLAND OF KNIGHTS AND KNAVES

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Randolph Martin
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1 CHPER 2 LOGIC. ON HE ISLND O KNIGHS ND KNVES Mathematics is not a spectator sport. If you want to improve, you have to play the sport! Do your homework! On separate quadrille paper hand it in at the beginning of the class. 2.1 Selected Review Homework 1. Using the distributive property rewrite (2x + y)(x y 2 4) = 2x(x y 2 4) + y(x y 2 4) = 5(3 4x + 3y)(x 10) = ( x + 5 3y)(x 10) = 5 3(x 10) 5 4x(x 10) + 5 3y(x 10) = 2. Compute (2n 1) Of course you can think of an algebraic solution: S n = (2n 3)+(2n 1) S n =(2n 1)+ (2n 3) S n =2n+ 2n+ + 2n+2n since there n terms S n = 2n n 2 = n 2 1

2 2.2 Classwork he problems in this section are related to a mythical island inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. hese problems come from books by Raymond M. Smullyan (he Lady or the iger?: and Other Logic Puzzles, What Is the Name of his ook?: he Riddle of Dracula and Other Logical Puzzles (Dover Recreational Math)). If you like these puzzles you can buy the books. 1. If oe is from that island can she say I am a knave? Our statements are : oe is a knight ; S oe is a knave hen S is the negation of. Knight= Knave= Who said What was said in oe S S he statement is made by us and has only one truth value according to our assumption. However, the statement S has two sources to get truth values: who said S, and what was said in S. Only when the two values agree, S is a valid statement. his cannot happen in either case, so an inhabitant from the island of knigths and knaves cannot make such a statement. 2. On the island of knights and knaves you meet two inhabitants Sue and ippy. Sue says: ippy is a knave. }{{} Statement I ippy says: } I and Sue {{ are knaves }. Statement II Solution S Sue is a knight ; ippy is a knight I ippy is a knave. hen I is the negation of. II ippy and Sue are knaves. hen II = not not S. Knight= Knave= Who said it What was said Sue ippy S:I :II I II 3. very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet two inhabitants: oey and Mel. oey tells you: Mell is a knave. Mel says Neither oey nor I are knaves. So who is a knight and who is a knave? Solution oey is a knight; M Mel is a knight S Mel is a knave. hen S is the negation of M. Neither oey nor I are knaves. We will learn next time to better analyze this kind of statement, but notice that is equivalent to saying that both and I have to be knights. 2

3 2.3 Solutions Classwork 1. he statements are very simple. oe is a knight ; S oe is a knave hen S is the negation of. Knight= Knave= Who said What was said in oe S S 2. In this problem the statements are more complex : S Sue is a knight ippy is a knight I ippy is a knave. hen I is the negation of. II ippy and Sue are knaves. hen II = not not S. Remember that I and II have two sources to get truth values: who said it, and what was said. Only the two values agree, I or II is a valid statement. he answer to this problem should make sure that both I and II are valid statements. Knight= Knave= Who said it What was said Sue ippy S:I :II I II 3. We first set up the following statements: M oey is a knight Mel is a knight S Mel is a knave. hen S is the negation of M. Neither oey nor I are knaves. We will learn next time to better analyze this kind of statement, but notice that is equivalent to saying that both and I have to be knights. Remember that S and have two sources to get truth values: who said it, and what was said. Only the two values agree, S or is a statement. he answer to this problem should make sure that both S and are statements. Knight= Knave= Who said it What was said oey Mel :S M: 3

4 2.4 Homework from the same island of knights and knaves Starred problems are optional. 1. You meet two inhabitants: Sally and ippy. Sally claims, I and ippy are not the same. ippy says, Of I and Sally, exactly one is a knight. Can you determine who is a knight and who is a knave? 2. You meet two inhabitants: Marge and oey. Marge says, oey and I are both knights or both knaves. oey claims, Marge and I are the same. Can you determine who is a knight and who is a knave? 3. You meet two inhabitants: Mel and ed. Mel tells you, Either ed is a knight or I am a knight. ed tells you that Mel is a knave. Can you determine who is a knight and who is a knave? 4. You meet two inhabitants: ed and eke. ed claims, eke could say that I am a knave. eke claims that it is not the case that ed is a knave. 5. You meet two inhabitants: ob and etty. ob claims that etty is a knave. etty tells you, I am a knight or ob is a knight. Can you determine who is a knight and who is a knave? 6. You meet two inhabitants: Carl and etty. Carl says, Neither etty nor I are knaves. etty claims, Carl and I are the same.can you determine who is a knight and who is a knave? 7. You meet two inhabitants: art and Mel. art claims, oth I am a knight and Mel is a knave. Mel tells you, I would tell you that art is a knight. Can you determine who is a knight and who is a knave? 8. You meet two inhabitants: ed and ippy. ed says, Of I and ippy, exactly one is a knight. ippy says that ed is a knave. Can you determine who is a knight and who is a knave? 9. You meet two inhabitants: Peggy and ippy. Peggy tells you that of ippy and I, exactly one is a knight. ippy tells you that only a knave would say that Peggy is a knave. Can you determine who is a knight and who is a knave? 10. lice, rian, and Charlie are from the island of knights and knaves. lice claims,?charlie could tell you that I am a knight.? rian says,?either lice is a knave, or I am a knight.? Charlie says that the others are either both knaves or both knights. What are lice, rian, and Charlie? 11. Now imagine that the island also have normals, who can either say truth or lie. my, ob, and Celine are from the island of knights, knaves, and normals. One of them is a knight, one is a knave, and one is normal. my says that Celine is a knave. ob says that my is a knight. Celine says that she is a normal. Can you figure out who is who? 4

5 2.5 few Elements of Logic Venn diagram is a representation with overlapping circles. he interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets. asic operations: Intersection of two sets, Union of two sets, Complement of in c = \ (i.e. what is in and not in ) bsolute complement of in the universe U c = U \ 5

Adaptive Fault Diagnosis using Self-Referential Reasoning by Robert Cowen

Adaptive Fault Diagnosis using Self-Referential Reasoning by Robert Cowen 1. Introduction. I first encountered logical puzzles about Knights, who always tell the truth, Knaves, who always lie, and Normals,