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The most common way of describing sets is by means of proposition forms.
3.24
Notation ()
Let
be a proposition form over a set
, and let
be a subset of
.
Then

(3.25) 
is defined to be the set of all elements
in
such that
is true.
The set described in (
3.25) is also written
In cases where the meaning of ``
'' is clear from the context, we may
abbreviate (
3.25) by
3.26
Examples.
is the set of all even integers, and
If and are sets, then
We will use the following notation throughout these notes.
3.30
Notation (
,
)
If
is an
integer we define
Thus
Similarly, if
is a real number, we define
3.31
Definition (Ordered pair.)
If
are objects, then the
ordered pair
is a
new object obtained by combining
. Two ordered
pairs
are equal if and only if
Similarly we may consider
ordered triples. Two ordered triples
are equal if and only if
We use the same notation
to represent an open interval in
and
an ordered pair in
. The context should always make it clear
which meaning is intended.
3.32
Definition (Cartesian product)
If
are sets then the
Cartesian product of and
is defined to be the set of all ordered pairs
such that

(3.33) 
3.34
Examples.
Let
be real numbers with
and
.
Then
and
Thus in general
.
The set is denoted by . You are familiar with one
Cartesian product. The euclidean plane
is the Cartesian
product of R with itself.
3.35
Exercise.
Let
and let
Sketch the sets
. For you should include an
explanation of how you arrived at your answer. For the other sets
no explanation is required.
3.36
Exercise.
Do there exist sets
such that
has
exactly five elements?
Next: 3.3 Functions
Up: 3. Propositions and Functions
Previous: 3.1 Propositions
Index
Ray Mayer
20070907