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74
Solutions to Exercises
17. Let
A
=
{a1,a2,
... ,an}
and
B
=
{b1,b2,
... ,b
m }.
Since
A
n
B
= 0,
ai
=f
bj
for any
i,j.
Let
C
=
{I, 2,
... ,
n
+
m}.
If
we can establish a onetoone correspondence between
A
U
B
and C, then
we will have
IA
U
BI
=
ICI
=
n
+
m. Define
f:
AU B
+
C
by
f(aI)
=
1,
f(a2)
=
2, .
.. ,
f(an)
=
n, f(b1)
=
n
+
1,
f(b2)
=
n
+
2, .
.. ,
f(b
m )
=
n
+
m. Since
f(x)
=
f(y)
+
x
=
y, f
is
onetoone. Clearly
f
is onto, so it is a onetoone correspondence.
18. The function
f:
(2,2)
+
(1,9) defined by
f(x)
=
2x
+
5 is a bijection.
1
1
19. Define
f:
(1,3)
+
R by
f(x)
=

2 and suppose
f(X1)
=
f(X2).
xI
1
1
1
1
1
1
Then
 
=


,
so

=
,
Xl

1
=
X2

1
and
Xl
=
X2.
Therefore,
f
Xl
 1
2
X2
 1
2
Xl
 1
X2
 1
.
G·
(0) I
2y
+
3
1S
onetoone.
lVen
y
E
,00,
et
X
=
2y
+
1·
We leave it to the student to verify that
f(x)
=
y.
Also 1
<
~::~
<
3. Finally, rng
f
=
(0,00), so
f
is a onetoone correspondence between (1,3) an.d (0,00).
20. (a) Assume to the contrary that
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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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