1. INTRODUCTION TO SETS
10
Sets can be written in a variety of ways. One can, of course, simply list the
elements if there are only a few. Another way is to use set builder notation, which
speciﬁes the sets using a predicate to indicate the attributes of the elements of the
set. For example, the set of even integers is
{
x

x
= 2
n,n
∈
Z
}
or
{
...,

2
,
0
,
2
,
4
,
6
,...
}
.
The ﬁrst set could be read as “the set of all x’s such that x is twice an integer.”
The symbol

stands for “such that.” A colon is often used for “such that” as well, so
the set of even integers could also be written
{
x
:
x
= 2
n,n
∈
Z
}
.
1.3. Common Universal Sets.
The following notation will be used throughout
these notes.
•
R
= the real numbers
•
N
= the natural numbers =
{
0
,
1
,
2
,
3
,...
}
•
Z
= the integers =
{
...,

3
,

2
,

1
,
0
,
1
,
2
,
3
,...
}
•
Z
+
= the positive integers =
{
1
,
2
,
3
,...
}
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