LECTURE 1: SETS, SUBSETS, AND SET OPERATIONS (
§
1.11.4)
There is surely a piece of divinity in us, something that was before the elements, and owes no homage
unto the sun. Sir Thomas Browne (1605  1682)
1.
What is a set?
A
set
is a collection of
elements
, where we don’t allow repetition and we ignore order.
Example 1.
Here are some examples of sets.
1. Let
S
=
{
1
,
2
,
3
}
=
{
1
,
3
,
2
}
=
{
3
,
2
,
1
}
. This is a set whose elements are 1, 2, and 3. We write
1
∈
S
, 2
∈
S
, and 3
∈
S
. Notice that 4
/
∈
S
. There are many ways to write down the same set.
2. Some sets get special names.
N
=
{
1
,
2
,
3
,...
}
= the natural numbers
Z
=
{
...,

2
,

1
,
0
,
1
,
2
}
= the integers
3. Some sets are described by properties:
T
=
{
1
,
2
,
3
,
4
,
5
}
=
{
x
∈
N
:
x
≤
5
}
.
U
=
{
1
,
4
,
9
,
16
}
=
{
x
∈
N
:
x
=
y
2
for some
y
∈
Z
}
=
{
x
2
:
x
∈
N
}
,
4. Here are ﬁve more important sets.
R
= the real numbers
C
= the complex numbers =
{
a
+
bi
:
a,b
∈
R
, i
2
=

1
}
Q
= the rational numbers =
{
a/b
:
∈
Z
, b
6
= 0
}
I
= the irrational numbers =
{
x
∈
R
:
x /
∈
Q
}
Where does
√
2 live? We have
√
2
∈
I
,
√
2
∈
R
,
√
2
∈
C
, but
√
2
/
∈
Q
.