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MATHEMATICAL INDUCTION
“Weak” Mathematical Induction
The principle of “weak” mathematical induction (“weak induction”, for
short) builds upon our most fundamental intuition about the structure of
the natural numbers: crudely put, that they are the result of starting with 0
and adding one, forever. Thus, if 0 has a certain property and if this prop
erty is also “inductive”, that is, roughly, if it is “passed on” from number to
number through the
add one
relation, it will follow that
every
natural number
has the property. This, in effect, is exactly the principle of weak induction;
a bit more formally put:
(
Weak Induction
) For any property
ϕ
of natural numbers: if 0
has
ϕ
and if, for any
n
∈
N
, from the assumption that
n
has
ϕ
it follows than
n
+
1 has
ϕ
, then all natural numbers have
ϕ
.
Let’s illustrate how the principle is used with a simple example.
Theorem.
For every natural number n
∈
N
,
0
+
1
+
...
+
n
=
n
(
n
+
1
)
2
.
Proof
: First we establish the
base case
, where we show that 0 has the property
in question — in this case the property
being a natural number n such that
0
+
1
+
...
+
n
=
n
(
n
+
1
)
2
.
And it is obvious that it does, as, when
n
=
0,
0
+
...
+
n
is just 0 and so we have:
(0.1)
0
=
0
2
=
0
(
1
)
2
=
0
(
0
+
1
)
2
Now we prove the
inductive case
. This is where we show that if an arbitrary
number
n
has the property in question, then so does its successor
n
+
1
.
So
assume that indeed the property is true of an arbitrary number
n
, that is,
that
(0.2)
0
+
1
+
...
+
n
=
n
(
n
+
1
)
2
This assumption is called the
induction assumption
or, more commonly, the
induction hypothesis
, socalled because it is the assumption we make in or
der to establish the inductive nature of our property. Given the induction
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View Full DocumentMATHEMATICAL INDUCTION
2
hypothesis (0.2), we need to show:
(0.3)
0
+
1
+
...
+
n
+ (
n
+
1
) =
(
n
+
1
)((
n
+
1
) +
1
)
2
.
But (0.3) follows from (0.2) almost immediately. For by (0.2), we have
(0.4)
0
+
1
+
...
+
n
+ (
n
+
1
) =
n
(
n
+
1
)
2
+ (
n
+
1
)
.
And by a series of simple inferences we have a chain of identies establishing
that
n
+
1 has the property in question:
n
(
n
+
1
)
2
+ (
n
+
1
)
=
n
(
n
+
1
)
2
+
2
(
n
+
1
)
2
(0.5)
=
n
(
n
+
1
) +
2
(
n
+
1
)
2
(0.6)
=
(
n
+
1
)(
n
+
2
)
2
(0.7)
=
(
n
+
1
)((
n
+
1
) +
1
)
2
.
(0.8)
Therefore, since we have shown that 0 has the property in question, and
that, if an arbitrary number
n
has it, so does its successor
n
+
1, we conclude
by the principle of weak induction that all numbers have the property in
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 Fall '09
 Yeelee

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