3. MATHEMATICAL INDUCTION
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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
.