3
The logical symbol
()
is the same as “if, and only if.” If
A
and
B
are any two statements,
then
A
()
B
is the same as saying
A
implies
B
and
B
implies
A
. It is also common to use
iff
in
this way.
4
Augustus De Morgan (1806–1871)
December 17, 2017
ª
lee/ira

4. FUNCTIONS AND RELATIONS
1-5
Two of the basic binary operations can be extended to work with indexed col-
lections. In particular, using the indexed collection from the previous paragraph,
we define
[
∏
2
§
A
∏
=
{
x
:
x
2
A
∏
for some
∏
2
§
}
and
\
∏
2
§
A
∏
=
{
x
:
x
2
A
∏
for all
∏
2
§
}.
De Morgan’s Laws can be generalized to indexed collections.
T
HEOREM
1.4.
If
{
B
∏
:
∏
2
§
}
is an indexed collection of sets and
A
is a set, then
A
\
[
∏
2
§
B
∏
=
\
∏
2
§
(
A
\
B
∏
)
and
A
\
\
∏
2
§
B
∏
=
[
∏
2
§
(
A
\
B
∏
).
P
ROOF
. The proof of this theorem is Exercise
1.4
.
⇤
4. Functions and Relations
4.1. Tuples.
When listing the elements of a set, the order in which they are
listed is unimportant; e.g.,
{
e
,
l
,
v
,
i
,
s
}
=
{
l
,
i
,
v
,
e
,
s
}
. If the order in which
n
items
are listed is important, the list is called an
n
-tuple
. (Strictly speaking, an
n
-tuple
is not a set.) We denote an
n
-tuple by enclosing the ordered list in parentheses.
For example, if
x
1
,
x
2
,
x
3
,
x
4
are four items, the 4-tuple (
x
1
,
x
2
,
x
3
,
x
4
) is different
from the 4-tuple (
x
2
,
x
1
,
x
3
,
x
4
).
Because they are used so often, the cases when
n
=
2 and
n
=
3 have special
names: 2-tuples are called
ordered pairs
and a 3-tuple is called an
ordered triple
.
D
EFINITION
1.5. Let
A
and
B
be sets. The set of all ordered pairs
A
£
B
=
{(
a
,
b
) :
a
2
A
^
b
2
B
}
is called the
Cartesian product
of
A
and
B
.
5
E
XAMPLE
1.2. If
A
=
{
a
,
b
,
c
} and
B
=
{1,2}, then
A
£
B
=
{(
a
,1),(
a
,2),(
b
,1),(
b
,2),(
c
,1),(
c
,2)}.
and
B
£
A
=
{(1,
a
),(1,
b
),(1,
c
),(2,
a
),(2,
b
),(2,
c
)}.
Notice that
A
£
B
6
=
B
£
A
because of the importance of
order
in the ordered pairs.
A useful way to visualize the Cartesian product of two sets is as a table. The
Cartesian product
A
£
B
from Example
1.2
is listed as the entries of the following
table.
£
1
2
a
(
a
,1)
(
a
,2)
b
(
b
,1)
(
b
,2)
c
(
c
,1)
(
c
,2)
5
René Descartes, 1596–1650
December 17, 2017
ª
lee/ira

1-6
CHAPTER 1. BASIC IDEAS
Of course, the common Cartesian plane from your analytic geometry course
is nothing more than a generalization of this idea of listing the elements of a
Cartesian product as a table.
The definition of Cartesian product can be extended to the case of more than
two sets. If {
A
1
,
A
2
,
···
,
A
n
} are sets, then
A
1
£
A
2
£
···
£
A
n
=
{(
a
1
,
a
2
,
···
,
a
n
) :
a
k
2
A
k
for 1
∑
k
∑
n
}
is a set of
n
-tuples. This is often written as
n
Y
k
=
1
A
k
=
A
1
£
A
2
£
···
£
A
n
.
4.2. Relations.
D
EFINITION
1.6. If
A
and
B
are sets, then any
R
Ω
A
£
B
is a
relation from
A
to B
. If (
a
,
b
)
2
R
, we write
aRb
.
In this case,
dom(
R
)
=
{
a
: (
a
,
b
)
2
R
}
Ω
A
is the
domain
of
R
and
ran(
R
)
=
{
b
: (
a
,
b
)
2
R
}
Ω
B
is the
range
of
R
. It may happen that
dom(
R
)
and
ran(
R
)
are proper subsets of
A
and
B
, respectively.

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