divided in half three times, and so on. The net effect of factors that are powers of two is
to defeat the simple argument that
f
grows “on average”.
40
CHAPTER 2. BASIC SET THEORY
Problem 2.46
True or false (and explain):
The
function
f
(
x
) =
x

1
x
+1
is a bijection from the real num
bers to the real numbers.
Problem 2.47
Find a function that is an injection
of the integers into the even integers that does not
appear in any of the examples in this chapter.
Problem 2.48
Suppose that
B
⊂
A
and that there
exists a bijection
f
:
A
→
B
. What may be reasonably
deduced about the set
A
?
Problem 2.49
Suppose that
A
and
B
are finite sets.
Prove that

A
×
B

=

A
 · 
B

.
Problem 2.50
Suppose that we define
h
:
N
→
N
as
follows. If
n
is even then
h
(
n
) =
n/
2
but if
n
is odd
then
h
(
n
) = 3
n
+1
. Determine if
h
is a (i) surjection
or (ii) injection.
Problem 2.51
Prove proposition 2.6.
Problem 2.52
Prove or disprove:
the composition
of injections is an injection.
Problem 2.53
Prove or disprove:
the composition
of surjections is a surjection.
Problem 2.54
Prove proposition 2.7.
Problem 2.55
List all permutations of
X
=
{
1
,
2
,
3
,
4
}
using oneline notation.
Problem 2.56
Suppose that
X
is a set and that
f
,
g
, and
h
are permutations of
X
. Prove that the equa
tion
f
◦
g
=
h
has a solution
g
for any given permu
tations
f
and
h
.
Problem 2.57
Examine the permutation
f
of
Q
=
{
a, b, c, d, e
}
which is
bcaed
in one line notation. If
we create the series
f, f
◦
f, f
◦
(
f
◦
f
)
, . . .
does the
identity function,
abcde
, ever appear in the series?
If so, what is its first appearance? If not, why not?
Problem 2.58
If
f
is a permutation of a finite set,
prove that the sequence
f, f
◦
f, f
◦
(
f
◦
f
)
, . . .
must
contain repeated elements.
Problem 2.59
Suppose that
X
and
Y
are finite sets
and that

X

=

Y

=
n
.
Prove that there are
n
!
bijections of
X
with
Y
.
Problem 2.60
Suppose that
X
and
Y
are sets with

X

=
n
,

Y

=
m
.
Count the number of functions
from
X
to
Y
.
Problem 2.61
Suppose that
X
and
Y
are sets with

X

=
n
,

Y

=
m
for
m > n
. Count the number of
injections of
X
into
Y
.
Problem 2.62
For a finite set
S
with a subset
T
prove that the permutations of
S
that have all mem
bers of
T
as fixed points form a set that is closed
under functional composition.
Problem 2.63
Compute
the
number
of
permuta
tions of a set
S
with
n
members that fix at least
m < n
points.
Problem 2.64
Using any technique at all, estimate
the fraction of permutations of an
n
element set that
have no fixed points. This problem is intended as an
exploration.
Problem 2.65
Let
X
be a finite set with

X

=
n
.
Let
C
=
X
×
X
.
How many subsets of
C
have the
property that every element of
X
appears once as a
first coordinate of some ordered pair and once as a
second coordinate of some ordered pair?
Problem 2.66
An alternate version of Sigma (
∑
)
and Pi (
producttext
) notation works by using a set as an index.
So if
S
=
{
1
,
3
,
5
,
7
}
then
summationdisplay
s
∈
S
s
= 16
and
productdisplay
s
∈
S
s
= 105
Given all the material so far, give and defend rea
sonable values for the sum and product of an empty
set.