Unit Fn
Functions
Section 1: Some Basic Terminology
Functions play a fundamental role in nearly all of mathematics. Combinatorics is no ex
ception. In this section we review the basic terminology and notation for functions. Per
mutations are special functions that arise in a variety of ways in combinatorics. Besides
studying them for their own interest, we’ll see them as a central tool in other topic areas.
Except for the real numbers
R
, rational numbers
Q
and integers
Z
, our sets are normally
finite. The set of the first
n
positive integers,
{
1
,
2
, . . . , n
}
will be denoted by
n
.
Recall that

A

is the number of elements in the set
A
. When it is convenient to do
so, we linearly order the elements of a set
A
.
In that case we denote the ordering by
a
1
, a
2
, . . . , a

A

or by (
a
1
, a
2
, . . . , a

A

). Unless clearly stated otherwise, the ordering on a
set of numbers is the numerical ordering. For example, the ordering on
n
is 1
,
2
,
3
, . . . , n
.
A review of the terminology concerning sets will be helpful.
When we speak about
sets, we usually have a “universal set”
U
in mind, to which the various sets of our discourse
belong. Let
U
be a set and let
A
and
B
be subsets of
U
.
•
The sets
A
∩
B
and
A
∪
B
are the
intersection
and
union
of
A
and
B
.
•
The set
A
\
B
or
A
−
B
is the
set difference
of
A
and
B
; that is, the set
{
x
:
x
∈
A, x
negationslash∈
B
}
.
•
The set
U
\
A
or
A
c
is the
complement
of
A
(relative to
U
). The complement of
A
is
also written
A
′
and
∼
A
.
•
The set
A
⊕
B
= (
A
\
B
)
∪
(
B
\
A
) is
symmetric difference
of
A
and
B
; that is, those
x
that are in
exactly one
of
A
and
B
. We have
A
⊕
B
= (
A
∪
B
)
\
(
A
∩
B
).
• P
(
A
) is the set of all subsets of
A
.
(The notation for
P
(
A
) varies from author to
author.)
• P
k
(
A
) the set of all subsets of
A
of size (or cardinality)
k
. (The notation for
P
k
(
A
)
varies from author to author.)
•
The Cartesian product
A
×
B
is the set of all ordered pairs built from
A
and
B
:
A
×
B
=
{
(
a, b
)

a
∈
A
and
b
∈
B
}
.
We also call
A
×
B
the
direct product
of
A
and
B
.
If
A
=
B
=
R
, the real numbers, then
R
×
R
, written
R
2
, is frequently interpreted as
coordinates of points in the plane. Two points are the same if and only if they have the
same coordinates, which says the same thing as our definition, (
a, b
) = (
a
′
, b
′
) if
a
=
a
′
and
b
=
b
′
. Recall that the direct product can be extended to any number of sets. How can
R
×
R
×
R
=
R
3
be interpreted?
Definition 1 (Function)
If
A
and
B
are sets, a
function
from
A
to
B
is a rule that tells
us how to find a
unique
b
∈
B
for each
a
∈
A
. We write
f
:
A
→
B
to indicate that
f
is a
function from
A
to
B
.
c
circlecopyrt
Edward A. Bender & S. Gill Williamson 2005. All rights reserved.