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CS 70
Discrete Mathematics for CS
Spring 2005
Clancy/Wagner
Notes 3
This lecture covers further variants of induction, including strong induction and the closely related well
ordering axiom. We then apply these techniques to prove properties of simple recursive programs.
Strong induction
Axiom 3.1 (Strong Induction)
: For any property
P
,
if
P
(
0
)
and
∀
n
∈
N
(
P
(
0
)
∧
P
(
1
)
∧
...
∧
P
(
n
) =
⇒
P
(
n
+
1
))
,
then
∀
n
∈
N
P
(
n
)
.
This says that if all the following sentences are true:
P
(
0
)
P
(
0
)
=
⇒
P
(
1
)
P
(
0
)
∧
P
(
1
)
=
⇒
P
(
2
)
P
(
0
)
∧
P
(
1
)
∧
P
(
2
)
=
⇒
P
(
3
)
P
(
0
)
∧
P
(
1
)
∧
P
(
2
)
∧
P
(
3
)
=
⇒
P
(
4
)
and so on, then
P
(
n
)
must be true for all
n
. Intuitively, this seems quite reasonable. If the truth of
P
all the
way up to
n
always implies the truth of
P
(
n
+
1
)
, then we immediately obtain the truth of
P
all the way up
to
n
+
1, which implies the truth of
P
(
n
+
2
)
, and so on
ad infinitum
.
If we compare the Strong Induction axiom to the original Induction axiom from Lecture Notes 2, we see that
Strong Induction appears to make it
easier
to prove things. With simple induction, one must prove
P
(
n
+
1
)
given the inductive hypothesis
P
(
n
)
; with strong induction one gets to assume the inductive hypothesis
P
(
0
)
∧
P
(
1
)
∧
...
∧
P
(
n
)
, which is much stronger.
Consider the following example, which is one half of the Fundamental Theorem of Arithmetic. (The other
half says that the product is unique.)
Theorem 3.1
:
Any natural number n
>
1
can be written as a product of primes.
To prove this, of course, we need to define prime numbers:
Definition 3.1 (Prime)
: A natural number
n
>
1 is prime iff it has exactly two factors (1 and
n
). 1 itself is
not prime.
Let’s see first what happens when we try a simple induction:
Proof
: (Attempt 1) The proof is by induction over the natural numbers
n
>
1.
• Base case: prove
P
(
2
)
.
P
(
2
)
is the proposition that 2 can be written as a product of primes. This is true, since 2 can be written
as the product of one prime, itself. (Remember that 1 is not prime!)
• Inductive step: prove
P
(
n
) =
⇒
P
(
n
+
1
)
for all natural numbers
n
>
1.
CS 70, Spring 2005, Notes 3
1
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View Full Document1. The inductive hypothesis states that
n
can be written as a product of primes.
2. To prove:
n
+
1 can be written as a product of primes.
3. We’re stuck: given
P
(
n
)
, we could easily establish
P
(
2
n
)
or
P
(
7
n
)
, but
P
(
n
+
1
)
is unconnected
to
P
(
n
)
.
2
With a strong induction, we can make the connection between
P
(
n
+
1
)
and earlier facts in the sequence that
are
relevant. For example, if
n
+
1
=
72, then
P
(
36
)
and
P
(
24
)
are useful facts.
Proof
: The proof is by strong induction over the natural numbers
n
>
1.
• Base case: prove
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