58
Solutions
to
Exercises
(b)
0
=
{x
E
R
I
f{x)
=
f{O)
=
O}
=
[0,1);
7/5
=
[1,2)
since L7/5J
=
1;
3/4
=
[1,0)
since
L
3/4J
=
1.
(c)
I
=
{I};
2
=
{2,
3,
6};
4
=
{4};
5
=
{5}.
28. (a) [BB] Here are the functions X
+
Y:
({a,
1),
{b,
In
({a,
1),
{b,
2n
({a,
1),
{b,
3n
({a,
2),
{b,
In
({a,
2),
{b,
2n
({a,
2),
{b,
3n
({a,
3),
{b,
In
({a,
3),
{b,
2n
({a,
3),
{b,
3n.
Here are the functions
Y
+
X:
{{I,a),{2,a),{3,a)}
{{I,a),{2,a),{3,b)}
({I,
a),
(2,
b),
{3,
an
({I,
a),
(2,
b),
{3,
bn
({I,
b),
(2,
a),
{3,
an
({I,
b),
(2,
a),
{3,
bn
({I,
b),
(2,
b),
{3,
an
({I,
b),
(2,
b),
{3,
bn.
(b) [BB] There are no onetoone functions
Y
+
X.
The onetoone functions from X
+
Y
are
({a,
1),
{b,
2n
({a,
1),
{b,
3n
({a,
2),
{b,
In
({a,
2),
{b,
3n
({a,
3),
{b,
In
({a,
3),
{b,
2n.
(c) [BB] There are no onto functions X
+
Y.
The onto functions
Y
+
X are
({I,
a),
(2,
a),
{3,
bn
({I,
a),
(2,
b),
{3,
an
({I,
a),
(2,
b),
{3,
bn
({I,
b),
(2,
a),
{3,
an
({I,
b),
(2,
a),
{3,
b)J
({I,
b),
(2,
b),
{3,
an.
29. (a) There are no onetoone functions
Y
+
X.
Here are the
24
onetoone functions X
+
Y.
({a,
1),
(b,
2),
{c,
3n
({a,
1),
(b,
2),
{c,
4n
({a,
1),
(b,
3),
{c,
2n
({a,
1),
(b,
3),
{c,
4n
({a,
1),
(b,
4),
{c,
2n
({a,
1),
(b,
4),
{c,
3n
({a,
2),
(b,
3),
{c,
In
({a,
2),
(b,
3),
{c,
4n
({a,
2),
(b,
4),
{c,
In
({a,
2),
(b,
4),
{c,
3n
({a,
2),
(b,
1),
(c,
3n
({a,
2),
(b,
1),
{c,
4n
({a,
3),
(b,
1),
{c,
2n
({a,
3),
(b,
1),
{c,
4n
({a,
4),
(b,
1),
{c,
2n
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 Summer '10
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 Graph Theory

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