Slide 1
Slide 1 text
cs2102: Discrete Mathematics
Class 3: Well-Ordering Principle
David Evans & Mohammad Mahmoody
University of Virginia
Slide 2
Slide 2 text
No content
Slide 3
Slide 3 text
There are lots of office hours!
(‘no Fridays’ is intentional)
Don’t wait to get help.
Slide 4
Slide 4 text
Plan
- Well-Ordering Principle
- Closer look: ordering / well ordering
- General Well-Ordering Theorem?
- Using Well-Ordering Principle
Slide 5
Slide 5 text
Well-Ordering Principle
Every nonempty set of non-negative integers has a smallest element.
Slide 6
Slide 6 text
Well-Ordering Principle
Every nonempty set of non-negative integers has a smallest element.
…is the crux of mathematical induction,
which is the basis for many of the most important proofs in CS.
Slide 7
Slide 7 text
What is an Ordered Set?
Definition. An ordered set :
(1) If ≠ then either < or <
(2) < and < then <
Slide 8
Slide 8 text
Examples of Ordered Sets
Slide 9
Slide 9 text
No content
Slide 10
Slide 10 text
No content
Slide 11
Slide 11 text
Well Ordered Sets
Definition. An ordered set (with respect to an
ordering `relation’ like < ) is well-ordered if any
of its non-empty subsets ⊆ has a minimum.
Slide 12
Slide 12 text
Well Ordering Principle (again)
Well Ordering Principle. The set of natural numbers
under the order “<“ is well ordered.
Every nonempty set of non-negative integers has a smallest element.
Why called “principle” ?
Slide 13
Slide 13 text
Examples of Well Ordered Sets
Slide 14
Slide 14 text
Well-Ordered?
Set: the integers Comparator: a < b
Slide 15
Slide 15 text
Well-Ordered?
Set: the integers Comparator: |a| < |b|
Slide 16
Slide 16 text
Well-Ordered?
Set: the non-negative reals Comparator: <
Slide 17
Slide 17 text
Well-Ordered?
Set: the non-negative rationals Comparator: <
Definition. A number is rational if it can be written as a ratio of two integers.
Slide 18
Slide 18 text
Set: the non-negative rationals Comparator: ?
Is there some comparator that makes the non-negative rationals well-ordered?
Slide 19
Slide 19 text
General Well-Ordering Theorem
For every set S, there is some ordering on that makes
well-ordered.
Is this true?
Slide 20
Slide 20 text
Slack break…
For every set S, there is some ordering on that makes
well-ordered.
Slide 21
Slide 21 text
Gyula Kőnig
There is no comparator
that results in a well-
ordering of the real
numbers.
Slide 22
Slide 22 text
Gyula Kőnig
Claimed counter-proof:
There is no comparator
that results in a well-
ordering of the real
numbers.
Proof withdrawn as incorrect
Slide 23
Slide 23 text
Georg Cantor
Well-ordering theorem is a
fundamental law of thought.
What should we (mathematics)
do when there is something
that seems like it should be
true, but no one can prove?
Slide 24
Slide 24 text
“General Well Ordering” Axiom?
Is it really that obvious?
Is it `equivalent’ to something else
that is obvious?
Slide 25
Slide 25 text
Axiom of Choice
For any collection of non-empty sets, there is a choice
function that picks exactly one element from each set.
Slide 26
Slide 26 text
No content
Slide 27
Slide 27 text
No content
Slide 28
Slide 28 text
Bertrand Russell
(1872-1970)
To choose one sock from
each of infinitely many
pairs of socks requires
the Axiom of Choice, but
for shoes the Axiom is
not needed.
Slide 29
Slide 29 text
For every set S, there is some ordering on that makes
well-ordered.
For any collection of non-empty sets, there is a choice
function that picks one element from each set.
implies
Well-Ordered Theorem
Axiom of Choice
Slide 30
Slide 30 text
For every set S, there is some ordering on that makes
well-ordered.
For any collection of non-empty sets, there is a choice
function that picks one element from each set.
Well-Ordered Theorem
Axiom of Choice
implies
Slide 31
Slide 31 text
Foundations of Math
ZF Axioms: Zermelo–Fraenkel Set Theory
ZFC Axioms: ZF + Axiom of Choice
Slide 32
Slide 32 text
Axiom of Choice: Still Controversial…
General Well Ordering Theorem on “larger sets”…
Banach-Tasrski Paradox
Well Ordering Principle (as stated) not controversial
Slide 33
Slide 33 text
How to Use Well Ordering Principle ?
Slide 34
Slide 34 text
Can I bet any dollar amount using just $2 and $5 chips?
Slide 35
Slide 35 text
Betable Numbers
A number is betable if it can be produced using
some combination of $2 and $5 chips.
Slide 36
Slide 36 text
Betable Numbers
A number is betable if it can be produced using
some combination of $2 and $5 chips.
Are all integers greater than $3 betable?
Slide 37
Slide 37 text
Betable Numbers
A number is betable if it can be produced using
some combination of $2 and $5 chips.
Are all integers greater than $3 betable?
Slide 38
Slide 38 text
Proof: Using Well-Ordering Principle
A number is betable if it can be produced using some
combination of $2 and $5 chips. All integers > 3 are betable.
Slide 39
Slide 39 text
Charge
• If you need to enroll in the class, bring us a
course action form to sign (now, or later)
• Due Friday (6:29pm): PS1
• Read MCS Ch 2, 3 (at least through 3.5) before
Thursday’s class