Lecture20-Review-Pf-by-Contradiction

Report
Peer Instruction in Discrete
Mathematics by Cynthia Leeis licensed
under a Creative Commons AttributionNonCommercial-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:
1.
2.
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
A.
B.
C.
D.
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
 What
A.
B.
C.
D.
person: “Everything I say is true.”
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
 What
A.
B.
C.
D.
person: “Everything I say is false.”
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]

A.
B.
C.
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

A.
B.
C.
D.
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.
We didn’t
Then A is a knave.
need this step
So what A says is false, and so there are zero knaves.
because we
But all three are knaves and zero are knaves is a contradiction.
had already
So B must be a knight, but we assumed B was a knave, a
reached a
contradiction.
contradiction.
So the assumption is false and the theorem is true. QED.

similar documents