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Section 5.1
121
the induction hypothesis). Since
x
rt
A k+1, x
belongs to an odd number of the sets
Al, A 2,
...
,Ak+1.
On the other hand, if
x
E
A
k
+
1
"
S,
then, since
x rt
s,
we know that
x
belongs to an even number
(possibly zero) of the sets
A 1, A 2, .
..
,Ak,
again by the induction hypothesis. Since
x
E
Ak+1'
we
conclude again that
x
belongs to an odd number of the sets
A 1, A 2, .
.. , Ak+1.
This proves that the
symmetric difference of
A
1 ,
.•. ,
A
k
+
1
consists of elements which belong to an odd number of these
sets.
Conversely, suppose
x
belongs to an odd number of the sets
A 1,
... ,
Ak+1.
We must show
x
belongs
to the symmetric difference of
A 1,
... ,
Ak+1.
If
x
belongs to
A k+1,
then
x
belongs to an even number
of the sets
A 1, .
.. , Ak
and, hence,
x
E
Ak+1
"
S
by the induction hypothesis. However, if
x
does
not belong to
Ak+lo
then clearly
xES"
A k+1.
Thus,
x
belongs to the symmetric difference of
Al, .
.. ,Ak+1.
By the Principle of Mathematical Induction, we conclude that for all
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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, Sets

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