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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '10
 any
 Graph Theory, Mathematical Induction, Sets, Natural number, odd number, induction hypothesis, Symmetric difference

Click to edit the document details