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Section 5.1
119
is true for
n
=
k.
Then
k+l
k
~)Xi
 XiI)
=
(~)Xi
 XiI))
+
(Xk+l

X(k+l)l)
i=2
=
(Xk
 Xl)
+
(Xk+l

Xk)
(by the induction hypothesis)
which is the fonnula with
n
=
k
+
1. Thus, the result holds for all
n
~
2 by the Principle of
Mathematical Induction.
11. [BB] The
k
to
k
+
1 step does not apply to the case
k
=
1. When
k
=
1,
G
=
{al, a2}.
Observe that
the groups
{al}, {a2}
have no member in common.
12. [BB] The induction was not started properly. When
n
=
1, the left side is 1, while the right side is £.
The statement is not true when
n
=
1.
13. As with the example at the end of this section, the problem is that the induction hypothesis has been
applied to an integer to which it does not necessarily apply. The induction hypothesis applies only to
integers
e
in the range 10
~
e
<
k,
10
~
e
being implicit because all integers in this problem are
at least
no
=
10. The induction hypothesis was applied to the integer
k

5, but this is not valid for
k
<
15.
14. The argument presented is flawed by its failure to specify
and to verify the statement for
n
=
no.
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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