3. MATHEMATICAL INDUCTION
83
3. Mathematical Induction
3.1. First Principle of Mathematical Induction.
Let
P
(
n
) be a predicate
with domain of discourse (over) the natural numbers
N
=
{
0
,
1
,
2
,...
}
. If
(1)
P
(0), and
(2)
P
(
n
)
→
P
(
n
+ 1)
then
∀
nP
(
n
).
Terminology: The hypothesis
P
(0) is called the
basis step
and the hypothesis,
P
(
n
)
→
P
(
n
+ 1), is called the
induction (or inductive) step
.
Discussion
The Principle of Mathematical Induction is an axiom of the system of natural
numbers that may be used to prove a quantiﬁed statement of the form
∀
nP
(
n
), where
the universe of discourse is the set of natural numbers. The principle of induction has
a number of equivalent forms and is based on the last of the four Peano Axioms we
alluded to in
Module 3.1 Introduction to Proofs
. The axiom of induction states that
if
S
is a set of natural numbers such that (i) 0
∈
S
and (ii) if
n
∈
S
, then
n
+ 1
∈
S
,
then
S
=
N
. This is a fairly complicated statement: Not only is it an “if .
.., then .
..”
statement, but its hypotheses
also
contains an “if .
.., then .
..” statement (if
n
∈
S
,
then
n
+ 1
∈
S
). When we apply the axiom to the truth set of a predicate
P
(
n
), we
arrive at the
ﬁrst principle of mathematical induction
stated above. More generally,
we may apply the principle of induction whenever the universe of discourse is a set of
integers of the form
{
k,k
+ 1
,k
+ 2
,...
}
where
k
is some ﬁxed integer. In this case
it would be stated as follows:
Let
P
(
n
) be a predicate over
{
k,k
+ 1
,k
+ 2
,k
+ 3
,...
}
, where
k
∈
Z
. If
(1)
P
(
k
), and
(2)
P
(
n
)
→
P
(
n
+ 1)
then
∀
nP
(
n
).
In this context the “for all
n
”, of course, means for all
n
≥
k
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document3. MATHEMATICAL INDUCTION
84
Remark
3.1.1
.
While the principle of induction is a very useful technique for
proving propositions about the natural numbers, it isn’t always necessary. There were
a number of examples of such statements in Module 3.2 Methods of Proof that were
proved without the use of mathematical induction.
Why does the principle of induction work? This is essentially the domino eﬀect.
Assume you have shown the premises. In other words you know
P
(0) is true and you
know that
P
(
n
) implies
P
(
n
+ 1) for any integer
n
≥
0.
Since you know
P
(0) from the basis step and
P
(0)
→
P
(1) from the inductive
step, we have
P
(1) (by
modus ponens
).
Since you now know
P
(1) and
P
(1)
→
P
(2) from the inductive step, you have
P
(2).
Since you now know
P
(2) and
P
(2)
→
P
(3) from the inductive step, you have
P
(3).
And so on ad inﬁnitum (or ad nauseum).
3.2. Using Mathematical Induction.
Steps
1. Prove the basis step.
2. Prove the inductive step
(a) Assume
P
(
n
) for arbitrary
n
in the universe. This is called the
induction
hypothesis
.
(b) Prove
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 PenelopeKirby
 Mathematical Induction, Natural number

Click to edit the document details