DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1982, 2008.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
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Chapter 3
THE NATURAL NUMBERS
3.1. Introduction
The set of natural numbers is usually given by
N
=
{
1
,
2
,
3
,...
}
.
However, this deﬁnition does not bring out some of the main properties of the set
N
in a natural way.
The following more complicated deﬁnition is therefore sometimes preferred.
Definition.
The set
N
of all natural numbers is deﬁned by the following four conditions:
(N1) 1
∈
N
.
(N2) If
n
∈
N
, then the number
n
+ 1, called the successor of
n
, also belongs to
N
.
(N3) Every
n
∈
N
other than 1 is the successor of some number in
N
.
(WO) Every nonempty subset of
N
has a least element.
The condition (WO) is called the Wellordering principle.
To explain the signiﬁcance of each of these four requirements, note that the conditions (N1) and
(N2) together imply that
N
contains 1
,
2
,
3
,....
However, these two conditions alone are insuﬃcient to
exclude from
N
numbers such as 5
.
5. Now, if
N
contained 5
.
5, then by condition (N3),
N
must also
contain 4
.
5
,
3
.
5
,
2
.
5
,
1
.
5
,
0
.
5
,

0
.
5
,

1
.
5
,

2
.
5
,...,
and so would not have a least element. We therefore
exclude this possibility by stipulating that
N
has a least element. This is achieved by the condition
(WO).
Chapter 3 : The Natural Numbers
page 1 of 6
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c
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W W L Chen, 1982, 2008
3.2. Induction
It can be shown that the condition (WO) implies the Principle of induction. The following two forms of
the Principle of induction are particularly useful.
PRINCIPLE OF INDUCTION (WEAK FORM).
Suppose that the statement
p
(
.
)
satisﬁes the
following conditions:
(PIW1)
p
(1)
is true; and
(PIW2)
p
(
n
+ 1)
is true whenever
p
(
n
)
is true.
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 Spring '08
 RAMASWAMY
 Math, Natural Numbers, Mathematical Induction, Natural number, Peano axioms, W W L Chen

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