1
CMPS 101
Algorithms and Abstract Data Types
Induction Proofs
Let
)
(
n
P
be a propositional function, i.e.
P
is a function whose domain is (some subset of) the set of
integers and whose codomain is the set {True, False}.
Informally, this means
)
(
n
P
is a sentence,
statement, or assertion whose truth or falsity depends on the integer
n
.
Mathematical Induction
is a proof
technique which can be used to prove statements of the form
)
(
:
0
n
P
n
n
≥
∀
(“for all
n
greater than or
equal to
0
n
,
)
(
n
P
is true”), where
0
n
is a fixed integer.
A proof by Mathematical Induction contains two
steps:
I.
Base Step:
Prove directly that the proposition
)
(
0
n
P
is true.
IIa.
Induction Step:
Prove
))
1
(
)
(
(
:
0
+
→
≥
∀
n
P
n
P
n
n
.
To do this pick an arbitrary
0
n
n
≥
, and assume for this
n
that
)
(
n
P
is true.
Then show as a
consequence that
)
1
(
+
n
P
is true.
The statement
)
(
n
P
is often called the
induction hypothesis
,
since it is what is assumed in the induction step.
When I and II are complete we conclude that
)
(
n
P
is true for all
0
n
n
≥
.
Induction is sometimes
explained in terms of a domino analogy.
Consider an infinite set of dominos which are lined up and ready
to fall.
Each domino is labeled by a positive integer, starting with
0
n
.
(It is often the case that
1
0
=
n
,
which we assume here for the sake of definiteness).
Let
)
(
n
P
be the assertion: “the
n
th domino falls”.
First prove
)
1
(
P
, i.e. “the first domino falls”, then prove
))
1
(
)
(
(
:
1
+
→
≥
∀
n
P
n
P
n
which says “if any
particular domino falls, then the next domino must also fall”.
When this is done we may conclude
)
(
:
1
n
P
n
≥
∀
, “all dominos fall”.
There are a number of variations on the induction step.
The first is just
a reparametrization of IIa.
IIb.
Induction Step:
Prove
))
(
)
1
(
(
:
0
n
P
n
P
n
n
→

>
∀
Let
0
n
n
>
, assume
)
1
(

n
P
is true, then prove
)
(
n
P
is true.
Forms
IIa
and
IIb
are said to be based on the
first principle of mathematical induction
.
The validity of
this principle is proved in the appendix of this handout.
Another important variation is called the
second
principle of mathematical induction
, or
strong induction
.
IIc.
Induction Step:
Prove
))
1
(
))
(
:
((
:
0
+
→
≤
∀
≥
∀
n
P
k
P
n
k
n
n
Let
0
n
n
≥
, assume for all
k
in the range
n
k
n
≤
≤
0
that
)
(
k
P
is true.
Then prove as a
consequence that
)
1
(
+
n
P
is true.
In this case the term
induction hypothesis
refers to the stronger
assumption:
)
(
:
k
P
n
k
≤
∀
.
The strong induction form is often reparametrized as in
IIb
:
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IId.
Induction Step:
Prove
))
(
))
(
:
((
:
0
n
P
k
P
n
k
n
n
→
<
∀
>
∀
Let
0
n
n
>
, assume for all
k
in the range
n
k
n
<
≤
0
, that
)
(
k
P
is true, then prove as a
consequence that
)
(
n
P
is true.
In this case the
induction hypothesis
is
)
(
:
k
P
n
k
<
∀
.
In terms of the Domino analogy, the strong induction form IId says we must show: (I) the first domino
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 Spring '09
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 Algorithms, Mathematical Induction, Natural number, induction step

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