Post on 30-Dec-2015
description
Well Ordering Principle
Every nonempty set of nonnegative integers has a least element
We actually used this already when arguing that a fraction can be reduced to “lowest terms”
The set of factors of a positive integer is nonempty
04/19/23
To prove P(n) for every nonnegative n:
• Let C = {n: P(n) is false} (the set of “counterexamples”)
• Assume C is nonempty in order to derive a contradiction
• Let m be the smallest element of C• Derive a contradiction (perhaps by
finding a smaller member of C)
04/19/23
A Proof Using WOP• Given a stack of pancakes, make a
nice stack with the smallest on top, then the next smallest, …, and the biggest on the bottom
• By using only one operation: Grabbing a wad off the top and flipping it!
• Theorem: n pancakes can be sorted using 2n-3 flips (n≥2)
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One way to do it• Grab under the
biggest pancake and bring it to the top
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• Flip the entire stack over
• Repeat, ignoring the bottom pancake
Why does this take 2n-3 flips?
• For n≥2, let P(n) := “n pancakes can be sorted using 2n-3 flips”
• Suppose this is false for some n• Let C = {n: P(n) is false}• C has a least element by WOP. Call it
m.• So m pancakes cannot be sorted using
2m-3 flips and m is the smallest number for which that is the case
04/19/23
Why does this take 2n-3 flips?
• m≠2 since one flip sorts 2 pancakes• But if m>2 then it takes 2 flips to get
the biggest pancake on the bottom …• and 2(m-1)-3 to sort the rest since
P(m-1) is true (since m-1 < m) …• for a total of 2(m-1)-3+2 = 2m-3,
contradicting the assumption that P(m) is false
04/19/23
Summing powers of 2
• Thm: For every n≥0, 1+2+22+23+…+2n =2n+1-1
• E.g. 1+2+22 = 1+2+4 = 7 = 23-1• Let P(n) be the statement
1+2+22+23+…+2n = 2n+1-1
04/19/23
Summing powers of 2
• Let C = {n: P(n) is false} = {n: 1+2+22+23+…+2n ≠2n+1-1}.
• Then C is nonempty by hypothesis.• Then C has a minimal element m by
WOP.• m cannot be 0 since P(0) is true:
1=20=20+1-1
• So m > 0
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Summing powers of 2
• But if1+2+22+23+…+2m ≠2m+1-1
• then subtracting 2m from both sides:1+2+22+23+…+2m-1 ≠2m+1-1-2m
= 2m-1(since 2m+2m = 2m+1)
• But then P(m-1) is also false, contradiction.
04/19/23