CS103
HO#18
SlidesInduction
4/15/11
1
The Principle of Mathematical Induction
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Suppose that:
We have a numbered collection of dominos
Each domino is standing
The first domino is knocked over
For any positive integer k, if domino k is
knocked over, it knocks over domino k + 1
Do all the dominos fall?
The Principle of Mathematical Induction
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Suppose that:
We have a numbered collection of dominos
Each domino is standing
The first domino is knocked over
For any positive integer k, if domino k is
knocked over, it knocks over domino k + 1
Let P
n
stand for the proposition: Domino n is knocked over.
We are given:
P
1
k(P
k
P
k+1
)
And we conclude
n P
n
The Principle of Mathematical Induction
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Be sure you understand the notation:
P
1
stands for the proposition: Domino 1 is knocked over.
P
2
stands for the proposition: Domino 2 is knocked over.
P
3
stands for the proposition: Domino 3 is knocked over.
P
4
stands for the proposition: Domino 4 is knocked over.
P
5
stands for the proposition: Domino 5 is knocked over.
...
P
k
stands for the proposition: Domino k is knocked over.
...
P
n
stands for the proposition: Domino n is knocked over.
...
P
n
:
You can reach rung n.
What would convince you that you
can reach all rungs of the ladder?
P
1
:
You can reach rung 1.
k(P
k
P
k+1
)
:
If you can reach
rung k, you can reach rung k + 1.
If we can prove those two statements,
then we can conclude:
n P
n
P
n
:
You can reach rung n.
What would convince you that you
can reach all rungs of the ladder?
P
1
:
You can reach rung 1.
k(P
k
P
k+1
)
:
If you can reach
rung k, you can reach rung k + 1.
P
n
:
You can reach rung n.
What would convince you that you
can reach all rungs of the ladder?
P
7
:
You can reach rung 7.
k(P
k
P
k+1
)
:
If you can reach
rung k, you can reach rung k + 1.
If we can prove those two statements,
then we can conclude:
n
7 (P
n
)
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CS103
HO#18
SlidesInduction
4/15/11
2
P
n
:
You can reach rung n.
What would convince you that you
can reach all rungs of the ladder?
P
2
:
You can reach rung 2.
k(P
k
P
k + 2
)
:
If you can reach
rung k, you can reach rung k + 2.
P
n
:
You can reach rung n.
What would convince you that you
can reach all rungs of the ladder?
P
2
:
You can reach rung 2.
k(P
k
P
k + 2
)
:
If you can reach
rung k, you can reach rung k + 2.
Instead
, we would renumber the rungs
we can reach 1, 2, ... then use ordinary
induction.
Or
, define R
k
= P
2k
and do induction
on the R's
The Principle of Mathematical Induction
A proof by mathematical induction that a proposition P
n
is true
for every positive integer n consists of two steps:
BASE CASE:
Show that P
1
is true.
INDUCTIVE STEP:
Assume that P
k
is true for an arbitrarily
chosen positive integer k and show that under that assumption
,
P
k+1
must be true.
From these two steps we conclude (by the principle of mathematical
induction) that for all positive integers n, P
n
is true.
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 Spring '11
 PLUMMER
 Mathematical Induction, Natural number, Prime number

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