Mathematics for Computer Science
MIT 6.042J/18.062J
The Well Ordering
Principle
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Albert R Meyer
September 12, 2011
Lec 2M.1
Well Ordering principle
Every nonempty set of
nonnegative integers
has a
least element.
Familiar? Now you mention it, Yes.
Obvious? Yes.
Trivial?
Yes. But watch out:
Albert R Meyer
September 12, 2011
Lec 2M.2
Well Ordering principle
Every nonempty set of
nonnegative integers
rationals
has a
least element.
NO!
Albert R Meyer
September 12, 2011
Lec 2M.3
Well Ordering principle
Every nonempty set of
nonnegative integers
has a
least element.
NO!
Albert R Meyer
September 12, 2011
Lec 2M.4
N ::= nonnegative integers
For rest of this talk,
“number” means
nonnegative integer
Albert R Meyer
September 12, 2011
Lec 2M.5
2
proof used Well Ordering
m
Proof: …suppose 2 
n
…can always find such m, n>0
without common factors…
why always ?
Albert R Meyer
September 12, 2011
Lec 2M.6
Proof using Well Ordering
Find smallest number m s.t.
m If m, n had a
2 .
n
common factor, c, then
m / c

2=
n / c 
Albert R Meyer
and m/c < m
September 12, 2011
Lec 2M.7
Proof using Well Ordering
Find smallest number m s.t.
m
2 .
n
This contradiction implies
m, n have no common factors.
Albert R Meyer
September 12, 2011
Lec 2M.8
What is the
• youngest age of MIT
graduate?
• smallest # neurons in
any animal?
• smallest #coins = $1.17?
Albert R Meyer
September 12, 2011
Lec 2M.9
Well Ordered Postage
available stamps:
5¢
3¢
n is postal if can make
(n+8)¢ postage from
3¢ & 5¢ stamps.
Albert R Meyer
September 12, 2011
Lec 2M.13
Well Ordered Postage
available stamps:
5¢
3¢
Thm: Every number is postal.
Prove by WOP. Suppose not.
Let m be least counterexample.
Albert R Meyer
September 12, 2011
Lec 2M.14
Well Ordered Postage
available stamps:
5¢
That is,
• m is not postal,
• any number < m is
postal.
Albert R Meyer
September 12, 2011
3¢
Lec 2M.15
Well Ordered Postage
0 is postal:
so m ≠ 0
Albert R Meyer
September 12, 2011
Lec 2M.16
Well Ordered Postage
m ≠ 1:
m ≠ 2:
Hence, m ≥ 3.
Albert R Meyer
September 12, 2011
Lec 2M.17
Well Ordered Postage
Now m-3 is a number < m,
so is postal. But then m
is postal too:
+
=
3¢ m+8¢
(m-3)+8¢ contradiction!
Albert R Meyer
September 12, 2011
Lec 2M.18
Geometric sums
Proof by WOP. Let m be
But = for
smallest n with .
n = 0, so m > 0, and
Albert R Meyer
September 12, 2011
Lec 2M.19
Geometric sums
Proof by WOP. Let m be
But = for
smallest n with .
n = 0, so m > 0, and
Albert R Meyer
September 12, 2011
Lec 2M.20
Geometric sums
add
m
r
to both sides
so = at m, contradicting :
there is no counterexample.
Albert R Meyer
September 12, 2011
Lec 2M.21
Well Ordering Principle Proofs
To prove
using WOP:
• define set of counterexamples
• assume C is not empty. By WOP,
have minimum element m ÎC
• Reach a contradiction somehow …
usually
c P(m)
ÎC with c < m
…orby
byfinding
proving
Albert R Meyer
September 12, 2011
Lec 2M.22
Team Problems
Problems
1−4
Albert R Meyer
September 12, 2011
Lec 2M.23