Section 5.1
127
39. (a) We prove that a set of
n
~
1 natural numbers has a smallest element.
If
n
=
1, this is true because
the single element of a set of size 1 is that set's smallest element. Now assume that
k
>
1 and that
the statement is true for all £
in
the range 1 ::; £ <
k.
We must prove it is true for
n
=
Consider
then a set
S
of
k
elements. Since
k
>
1, we can remove a single element
a
from
S.
The remaining
k

1 elements have a smallest element
b
(by the induction hypothesis) and the smaller of
a
and
b
is the smallest element of
This proves that any nonempty
finite
set of natural numbers has a
smallest element.
It
then follows by Exercise 16 that
any
nonempty set of natural numbers has a
smallest element.
(b) Given a statement
P
involving the integer
n,
suppose
1.
P
is true for
n
=
no,
and
2. if
k
>
no
and
P
is true for all
n
in the range
::;
n
<
k,
then it's true also for
We must prove
P
is true for all integers
n
~
no.
Proof.
P
is not true for all integers
n
~
then the set
S
of such integers is a nonempty set
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 Summer '10
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 Graph Theory, Natural Numbers, Natural number, smallest element, IAL

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