Section 1.1:
Preliminaries  Sets, Functions, and Induction
In this section, we deal with a few preliminaries. Having a strong foundation will make the rest of the text much
easier to digest. First, we discuss some basic set theory. We then discuss functions and some special types of
functions. We end with a discussion on proofs by induction.
BASICS of SET THEORY
Set theory is the language of mathematics. In order for us to effectively communicate theorems and definitions,
we require at least a naïve understanding of this topic. Here, the term “naïve” refers to the fact that many of our
arguments will involve words including “and,” “or,” “for some,” and “for every” without a rigorous definition
of what these terms mean.
Definition:
A
set
S
is a collection of objects. The objects in a set are called its
elements
. If
x
is an element of a
set
S
, we write
x
S
∈
; otherwise we write
x
S
∉
.
We have two conventional ways of denoting a set. Either we list all of its elements between curly brackets, e.g.
{
}
1,2,3
, or we write a rule for describing the elements of the set, also in curly brackets, e.g.
{
}
1
3 and
is an integer
x
x
x
≤
≤
where the vertical bar is read “such that.” Notice that the two examples
actually describe the same set. Also note that the order in which terms appear in a set do not matter, e.g. .
Definition:
Two sets
A
and
B
are said to be
equal
if and only if they contain exactly the same elements. If
A
and
B
are equal, we write
A
B
=
; otherwise we write
A
B
≠
.
Definition:
Let
A
and
B
be sets. We say that
A
is a
subset
of
B
, or that
A
is
contained
in
B
, if every element of
A
is also an element of
B
, that is, if
x
A
∈
, then
x
B
∈
also. If this is the case, we write
A
B
⊂
; otherwise, we
write
A
B
⊄
. We consider any set a subset of itself. If
A
B
⊂
with
A
nonempty (see below) and
A
B
≠
,
then
A
is called a
proper
subset of
B
.
Example:
Let
{
}
1,3,5,7
A
=
and let
{
}
1,2,3,4,5,6,7,8,9
B
=
. Then clearly
A
B
⊂
but
B
A
⊄
.
Example:
Let
{
}
1,1
A
= 
and let
{
}
2

1
0
B
x x
=
 =
. It follows that
A
B
⊂
and
B
A
⊂
. In fact, we find that
A
B
=
.
Theorem:
Let
A
and
B
be sets. Then
A
B
=
if and only if
A
B
⊂
and
B
A
⊂
.
Proof
Suppose
A
B
=
. Then by definition,
A
and
B
contain exactly the same elements. Consider some
x
A
∈
.
Since
A
and
B
contain exactly the same elements, it follows that
x
B
∈
and therefore,
A
B
⊂
. A similar
argument shows that
B
A
⊂
.
Conversely, suppose that
A
B
⊂
and
B
A
⊂
. Then for every
x
A
∈
, it follows that
x
B
∈
; similarly, for
every
x
B
∈
, we have
x
A
∈
. While this in itself would be enough to prove the statement, let us argue by
contradiction. Assume that
A
and
B
do not have exactly the same elements. Then either
A
has an element that is
not in
B
or
B
has an element that is not in
A
. Suppose
without loss of generality
that there is an element of
A
that
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not
an element of
B
. This contradicts that
A
B
⊂
and so our initial assumption that
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 Spring '08
 PLOTKIN
 Set Theory, Sets, Natural number, positive integer

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