Math 100
Functions
Martin H. Weissman
1.
Definitions
The concept of “function” is a core concept in the highschool curriculum.
There are many ways that
functions are taught at the highschool level:
•
Functions are graphs which pass the vertical line test.
•
Functions are “machines” which output one number for each number input.
•
Functions are expressions involving a bunch of operations and constants, and one letter, called
x
.
In advanced mathematics, we use a definition of “function” which is based on set theory.
The abstract
definition is most closely related to the first highschool definition – the vertical line test – though the
abstract definition is less obviously geometric.
Let
A
and
B
be sets. Then
A
×
B
(called the Cartesian product of
A
and
B
) is the set of ordered pairs, whose
first entry is an element of
A
, and whose second entry is an element of
B
. For example, if
A
=
{
a, b, c
}
, and
B
=
{
a, d
}
, then the set
A
×
B
has the following six elements:
A
×
B
=
{
(
a, a
)
,
(
a, d
)
,
(
b, a
)
,
(
b, d
)
,
(
c, a
)
,
(
c, d
)
}
.
In general, if
A
and
B
are finite sets, and
A
has
x
elements, and
B
has
y
elements, then
A
×
B
is a set with
x
×
y
elements. Hopefully, this justifies the notation!
Definition 1.
A function from
A
to
B
is a subset
f
⊂
A
×
B
, such that the following condition holds:
•
For all
a
∈
A
, there exists
b
∈
B
, such that (
a, b
)
∈
f
.
•
For all
a
∈
A
, for all
b
∈
B
, and for all
b
0
∈
B
, if (
a, b
)
∈
f
and (
a, b
0
)
∈
f
, then
b
=
b
0
.
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 Fall '08
 Weissman
 Math, Finite set, Martin H. Weissman

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