Mathematical Induction
Victor Adamchik
Fall of 2005
Lecture 1 (out of three)
°
Plan
1. The Principle of Mathematical Induction
2. Induction Examples
°
The Principle of Mathematical Induction
Suppose we have some statement
P
°
n
±
and we want to demonstrate that
P
°
n
±
is true for all
n
°
°
. Even if we can provide proofs for
P
°
0
±
,
P
°
1
±
,
...,
P
°
k
±
, where
k
is some large
number, we have accomplished very little. However, there is a general method, the
Princi
ple of Mathematical Induction
.
Induction is a defining difference between discrete and continuous mathematics.
Principle of Induction
.
In order to show that
±
n
,
P
°
n
±
holds, it suffices to establish the
following two properties:
(I1)
Base case
: Show that
P
°
0
±
holds.
(I2)
Induction step
: Assume that
P
°
n
±
holds, and show that
P
°
n
²
1
±
also holds.
In the induction step, the assumption that
P
°
n
±
holds is called the
Induction Hypothesis
(IH). In more formal notation, this proof technique can be stated as
²
P
°
0
± ³
±
k
°
P
°
k
±
³
P
°
k
²
1
±±´
´±
n P
°
n
±
V. Adamchik
21127: Concepts of Mathematics
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You can think of the proof by (mathematical) induction as a kind of recursive proof:
Instead of attacking the problem directly, we only explain how to get a proof for P
°
n
²
1
±
out of a proof for P
°
n
±
.
How would you prove that the proof by induction indeed works??
Proof
(by contradiction) Assume that for some values of
n
,
P
°
n
±
is false. Let
n
0
be the least
such
n
that
P
°
n
0
±
is false.
n
0
cannot be 0, because
P
°
0
±
is true. Thus,
n
0
must be in the form
n
0
µ
1
²
n
1
. Since
n
1
¶
n
0
then by
P
°
n
1
±
is true. Therefore, by inductive hypothesis
P
°
n
1
²
1
±
must be true. It follows then that
P
°
n
0
±
is true.
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 Spring '10
 RazaSuleman
 Mathematical Induction, Recursion, Inductive Reasoning, Natural number, Mathematical logic, Mathematical proof

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