Section 3.2
n
1
2
3
m
1
1
2
3
2
1
4
9
30. [BB]
3
1
8
27
4
1
16
81
31. (a) LetA
= {al,a2,
...
,a
n }.
4
4
16
64
256
We guess that the number of
functions
X
+
Y
is
n
m.
59
If
I: A
+
B
were onetoone, then ICal),
l(a2)'
... ' I(an)
would be
n
different elements in
B
contradicting the fact that Bhas only
m
<
n
elements.
(b) Let
B
=
{bit b2,
... ,
bm}.
If
I: A :tB
were onto, then there would exist elements alt
a2,
... ,
am
in
A
such that
I(ai)
==
bi,
distinct
by~efinition
of "function." But this contradicts the fact that
A
has
n
<
m
elements.
32. (a) [BB
(+)]
Suppose
A
and
B
each contain n.elements. Assume that
I:
A
+
B
is onetoone
and let
C
=
{I(a)
I
a
E
A}.
Since
leal)
=/:
1(0.2)
if al
=/:
a2, C
is a subset of
B
containing
n
eiements; so
C
=
B.
Therefore,
I
is onto.
Conversely, suppose
I
is onto. Then
{I
(a)
I
a
E
A}
=
B
and so the set on the left here contains
n
distinct elements. Since Acontains only
n
elements, we cannot have
I(al)
=
l(a2)
for distinct
ai,
a2;
thus,
I
is onetoone.
(b)
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 Graph Theory, elements, lt a2

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