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Section 3.2
65
25. [BB] Since a bijective function is, by definition, a onetoone onto function, we conclude, by the results
of part (a) of the previous two exercises, that indeed the composition of bijective functions is bijective.
26.
(a)
f(1000)
=
998;
f(999)
=
fU(1003))
=
f(1001)
=
999;
f(998)
=
fU(1002))
=
f(1000)
=
998; f(997)
=
fU(1001))
=
f(999)
=
999.
(b) We guess that
f(n)
=. .
.
{
998
if
n
is even
999
If
n
IS odd.
(c) We guess rng
f
=
{998,999}.
2
2
()
()
Xl
X2
Xl
X2
2 2
2
2 2
27. Suppose
f Xl
=
f X2
. Then
~
=
~,so
~2
=
~2'
XIX2
+
2XI
=
XIX2 +
Y xi
+
2
y
x~
+
2
xl
+
x
2
+
2x~,
2xi
=
2x2
and Xl
=
±X2.
Note that
X2
= Xl
is not possible (unless Xl
=
X2
=
0) since
Xl
Xl
.
f(
Xl)
=
~
=I
~
=
f(XI)'
Thus
f
IS onetoone. Next we note that
yxi
+
2
yxi
+
2
X
Y
E
rng
f
~
Y
=
f (x)
for some
X
E R
~
Y
= .y'X"2+2
for some
X
E
R.
X2
+2
Now, if
y
=
~,
then
y2(X2
+
2)
=
X2,
so
x2(y2

1)
=
_2y2
and this implies that
y2 =I
1
X2
+2
(otherwise we have the equation 0
=
1)
and also
y2
'1 :::;
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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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