Math 117 – April 14, 2011
6
Natural and real numbers
The set of natural numbers is
N
=
{
1
,
2
,
3
,
4
, . . .
}
. We will assume that the operations of addition (+)
and multiplication (
·
), as well as the relation “
≤
” are well defined on the set of real numbers, and
hence on the set of natural numbers as well. We further assume that
N
satisfies the following property.
Axiom 6.1
(Wellordering property)
.
If
S
⊆
N
is a nonempty subset, then there exists an element
m
∈
S
, such that
m
≤
k
for all
k
∈
S
.
The principle of mathematical induction is a consequence of the wellordering property, and is a useful
tool when proving theorems about natural numbers.
Theorem 6.2
(Principle of mathematical induction)
.
Let
P
(
n
)
be a statement for every
n
∈
N
. Then
P
(
n
)
is true for all
n
∈
N
, provided
(
a
)
P
(1)
is true (basis for the induction)
(
b
)
∀
k
∈
N
, if
P
(
k
)
is true, then
P
(
k
+ 1)
is true (inductive step).
A generalization of the math induction is the principle of strong induction.
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 Spring '08
 Akhmedov,A
 Real Numbers, Addition, Multiplication, Natural Numbers, Natural number, ∀x

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