set
A set is a collection of distinct objects which we call elements of the set. A set can have a finite number of elements or infinitely many elements.
declaration of a finite set
We may declare a finite
set by listing it's elements between curly braces like so
or
{
1
,
2
,
3
,
41
}
Suppose we wrote out the set
{
a
,
b
,
c
,
d
}
and then later on realized that
a
=
b
=
c
=
d
, then this set is simply
{
a
,
a
,
a
,
a
}
, which we define to just be
{
a
}
. Therefore we allow duplicates when declaring a set, but realize the set it generates has no such duplicate, so
{
1
,
1
,
1
}
=
{
1
,
1
}
=
{
1
}
set builder notation
Suppose that
P
(
x
)
represents a statement which depends on the variable
x
, then we define
{
x
∈
S
:
P
(
x
)
}
to be the set of elements of
S
such that the statement
P
holds true.
Empty Set
The
set with no elements is called the empty set and is denoted by
∅
element of
Suppose that
a
is an element of a
set
S
, then we write
a
∈
S
set equality
Given two sets
A
,
B
, we say that
A
and
B
are equal and write
A
=
B
when
x
∈
A
if and only if
x
∈
B
Family of Sets
We denote a
set whose elements are also sets by a family of sets to avoid the unwieldy statement: "set of sets"
subset
Given the sets
A
,
B
, we say that
A
is a subset of
B
when for every
a
∈
A
,
a
is also in
B
. When this is the case we write
A
⊆
B
superset
Given the sets
A
,
B
, we say that
A
is a superset of
B
when for every
b
∈
B
,
b
is also in
A
. When this is the case we write
A
⊇
B
set equality by subsets
Suppose that
A
,
B
are sets such that
A
⊆
B
and
B
⊆
A
, then
A
=
B
We will show the sets are equal using the definition of set equality.
To show that a bi-implication is true, we have to prove both directions. So first suppose that
x
∈
A
, then since
A
⊆
B
,
x
∈
B
. For the other direction suppose that
x
∈
B
, then since
B
⊆
A
then
x
∈
A
. Therefore we know that
x
∈
A
if and only if
x
∈
B
as needed.
rationals
The integers are the
set
Q
:=
{
p
q
:
p
,
q
∈
Z
,
q
≠
0
}
note: The symbol for rationals can be remebered as they are
Q
uotients
integers
The integers are the
set
Z
:=
{
…
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
}
note: The symbol for integers can be remembered as the word for number in German is
Z
ahlen
Natural Numbers 0
We define
N
0
:=
{
0
,
1
,
2
,
3
,
4
,
…
}
Finite Integer Set
Suppose that
a
,
b
∈
Z
, then the notation
{
a
,
…
b
}
is a
set
X
such that for any
x
∈
Z
if
a
≤
x
≤
b
then
x
∈
X
.
Empty Integer Set
Suppose that
a
,
b
∈
Z
such that
a
>
b
then
{
a
,
…
b
}
=
∅
{
a
,
…
b
}
is
defined as the set
X
such that
x
∈
X
iff
x
∈
Z
and
a
≤
x
≤
b
but this is impossible for if there were any such
x
then by transitivity we have that
a
≤
b
which is impossible.
Natural Numbers 1
We define
N
1
:=
{
1
,
2
,
3
,
4
,
…
}
Natural Numbers up to n
Let
n
∈
N
1
, then we define
[
n
]
:=
{
1
,
2
,
…
,
n
}
power set
Given a
set
X
we define the power set of
subsets
X
as
P
(
X
)
:=
{
S
:
S
⊆
X
}