CHAPTER
2
LOGIC. ON THE ISLAND OF KNIGHTS AND KNAVES
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) = (5 · 3 − 5 · 4x + 5 · 3y)(x − 10) = 5 · 3(x − 10) − 5 · 4x(x − 10) + 5 · 3y(x − 10) = · · ·
2. Compute 1 + 3 + 5 + 7 + 9 + · · · (2n − 1)
Of course you can think of an algebraic solution:
Sn
=1+
Sn
=(2n − 1)+
2Sn
=2n+
3+ · · · +
(2n − 3)+ · · · +
2n+ · · · +
since there n terms
Sn =
2n · n
= n2
2
1
(2n − 3)+(2n − 1)
3+1
2n+2n
2.2
Classwork
The 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. These problems come from books by Raymond M. Smullyan
(The Lady or the Tiger?: and Other Logic Puzzles, What Is the Name of This Book?: The Riddle of Dracula
and Other Logical Puzzles (Dover Recreational Math)). If you like these puzzles you can buy the books.
1. If Zoe is from that island can she say ”I am a knave” ?
Our statements are :
Z ”Zoe is a knight” ; S ”Zoe is a knave” Then S is the negation of Z.
Knight=T
Knave=F
Who said
What was said in
Zoe
S
S
The statement Z is made by us and has only one truth
T
F
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. This 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 Zippy. Sue says: ”Zippy is a knave”.
{z
}
|
Statement I
Zippy says: ”I
| and Sue{zare knaves”}.
Statement II
Solution
S Sue is a knight ; Z Zippy is a knight
I Zippy is a knave. Then I is the negation of Z.
II Zippy and Sue are knaves. Then II = notZ ∧ not S.
Knight=T
Knave=F
Sue Zippy
T
T
F
F
Who said it
S:I
Z:II
What was said
I
II
T
F
T
F
3. A very special island is inhabited only by knights and knaves. Knights always tell the truth, and
knaves always lie. You meet two inhabitants: Zoey and Mel. Zoey tells you: ”Mell is a knave”. Mel
says ”Neither Zoey nor I are knaves”. So who is a knight and who is a knave ?
Solution
Z Zoey is a knight; M Mel is a knight
S Mel is a knave. Then S is the negation of M.
T Neither Zoey nor I are knaves.
We will learn next time to better analyze this kind of statement, but notice that T is equivalent
to saying that both Z and I have to be knights.
2
2.3
Solutions Classwork
1. The statements are very simple. Z ”Zoe is a knight” ; S ”Zoe is a knave” Then S is the negation of
Z.
Knight=T
Knave=F
Who said What was said in
Zoe
S
S
T
F
T
F
F
T
2. In this problem the statements are more complex :
S Sue is a knight
Z Zippy is a knight
I Zippy is a knave. Then I is the negation of Z.
II Zippy and Sue are knaves. Then II = notZ ∧ 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. The answer to this problem should make sure that
both I and II are valid statements.
Knight=T
Knave=F
Sue Zippy
T
T
F
F
T
F
T
F
Who said it
S:I
Z:II
T
T
F
F
T
F
T
F
What was said
I
II
F
T
F
T
F
F
3. We first set up the following statements:
Z Zoey is a knight
M Mel is a knight
S Mel is a knave. Then S is the negation of M.
T Neither Zoey nor I are knaves.
We will learn next time to better analyze this kind of statement, but notice that T is equivalent
to saying that both Z and I have to be knights.
Remember that S and T have two sources to get truth values: who said it, and what was said. Only
the two values agree, S or T is a statement. The answer to this problem should make sure that both
S and T are statements.
Knight=T
Knave=F
Zoey Mel
T
T
F
F
T
F
T
F
Who said it
Z:S M:T
T
T
F
F
T
F
T
F
What was said
F
T
F
T
F
F
3
2.4
Homework from the same island of knights and knaves
Starred problems are optional.
1. You meet two inhabitants: Sally and Zippy. Sally claims, ”I and Zippy are not the same.” Zippy 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 Zoey. Marge says, ”Zoey and I are both knights or both knaves.”
Zoey 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 Ted. Mel tells you, ”Either Ted is a knight or I am a knight.” Ted
tells you that Mel is a knave. Can you determine who is a knight and who is a knave?
4. You meet two inhabitants: Ted and Zeke. Ted claims, ”Zeke could say that I am a knave.” Zeke claims
that it is not the case that Ted is a knave.
5. You meet two inhabitants: Bob and Betty. Bob claims that Betty is a knave. Betty tells you, ”I am a
knight or Bob is a knight.” Can you determine who is a knight and who is a knave?
6. You meet two inhabitants: Carl and Betty. Carl says, ”Neither Betty nor I are knaves”. Betty claims,
”Carl and I are the same”.Can you determine who is a knight and who is a knave?
7.∗ You meet two inhabitants: Bart and Mel. Bart claims, ”Both I am a knight and Mel is a knave”. Mel
tells you, ”I would tell you that Bart is a knight”. Can you determine who is a knight and who is a
knave?
8.∗ You meet two inhabitants: Ted and Zippy. Ted says, ”Of I and Zippy, exactly one is a knight.”Zippy
says that Ted is a knave. Can you determine who is a knight and who is a knave?
9.∗ You meet two inhabitants: Peggy and Zippy. Peggy tells you that ”of Zippy and I, exactly one is a
knight”. Zippy 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.∗ Alice, Brian, and Charlie are from the island of knights and knaves. Alice claims, ?Charlie could tell
you that I am a knight.? Brian says, ?Either Alice is a knave, or I am a knight.? Charlie says that the
others are either both knaves or both knights. What are Alice, Brian, and Charlie?
11.∗ Now imagine that the island also have normals, who can either say truth or lie. Amy, Bob, 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. Amy says that Celine is a knave. Bob says that Amy is a knight. Celine says that she is a
normal. Can you figure out who is who?
4
2.5
A few Elements of Logic
A Venn diagram is a representation with overlapping circles. The 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.
Basic operations: Intersection of two sets A ∩ B, Union of two sets A ∪ B, Complement of ”B” in ”A”
B c ∩ A = A \ B(i.e. ”what is in A and not in B”) Absolute complement of A in the universe U Ac = U \ A
B
A
B
A
5
A
B