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The most common way of describing sets is by means of proposition forms.
3.24
Notation (
)
Let
![$P$](img550.gif)
be a proposition form over a set
![$S$](img49.gif)
, and let
![$T$](img104.gif)
be a subset of
![$S$](img49.gif)
.
Then
![\begin{displaymath}
\{x: x\in T \mbox{ and } P(x) \}
\end{displaymath}](img637.gif) |
(3.25) |
is defined to be the set of all elements
![$x$](img35.gif)
in
![$T$](img104.gif)
such that
![$P(x)$](img563.gif)
is true.
The set described in (
3.25) is also written
In cases where the meaning of ``
![$T$](img104.gif)
'' 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
![$n$](img9.gif)
is an
integer we define
Thus
Similarly, if
![$a$](img31.gif)
is a real number, we define
3.31
Definition (Ordered pair.)
If
![$a,b$](img152.gif)
are objects, then the
ordered pair ![$(a,b)$](img653.gif)
is a
new object obtained by combining
![$a \mbox{ and } b$](img654.gif)
. Two ordered
pairs
![$(a,b) \mbox{ and } (c,d)$](img655.gif)
are equal if and only if
![$ a = c \mbox{ and } b = d.$](img656.gif)
Similarly we may consider
ordered triples. Two ordered triples
![$(a,b,x) \mbox{ and } (c,d,y)$](img657.gif)
are equal if and only if
![$a = c \mbox{ and } b = d
\mbox{ and } x = y.$](img658.gif)
We use the same notation
![$(a,b)$](img653.gif)
to represent an open interval in
![$\mbox{{\bf R}}$](img153.gif)
and
an ordered pair in
![$\mbox{{\bf R}}^2$](img169.gif)
. The context should always make it clear
which meaning is intended.
3.32
Definition (Cartesian product)
If
![$A,B$](img659.gif)
are sets then the
Cartesian product of
and ![$B$](img145.gif)
is defined to be the set of all ordered pairs
![$(x,y)$](img94.gif)
such that
![\begin{displaymath}
A \times B = \{ (x,y) : x \in A \mbox{ and } y \in B \}\end{displaymath}](img661.gif) |
(3.33) |
3.34
Examples.
Let
![$a, b, c, d$](img112.gif)
be real numbers with
![$a\leq b$](img113.gif)
and
![$c\leq d$](img114.gif)
.
Then
and
Thus in general
![$A \times B \not= B \times A$](img664.gif)
.
The set
is denoted by
. You are familiar with one
Cartesian product. The euclidean plane
is the Cartesian
product of R with itself.
3.35
Exercise.
Let
![$S = B(-2,2:-2,2)$](img667.gif)
and let
Sketch the sets
. For
you should include an
explanation of how you arrived at your answer. For the other sets
no explanation is required.
3.36
Exercise.
Do there exist sets
![$A,B$](img659.gif)
such that
![$A \times B$](img674.gif)
has
exactly five elements?
Next: 3.3 Functions
Up: 3. Propositions and Functions
Previous: 3.1 Propositions
  Index
Ray Mayer
2007-09-07