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3.37
Definition (Function.)
Let
be sets. A
function with domain and codomain is
an ordered triple
where
is a rule which assigns to each
element of
a unique element of
.
The element of
which
assigns to an element
of
is denoted
by
. We call
the
-image of or the
image of under .
The notation
is an abbreviation for ``
is a function
with domain
and codomain
''. We read ``
'' as
``
is a function from
to
.''
3.38
Examples.
Let
be defined by the rule
Then
is not defined, because
Let
be defined by the rule: for all
Thus
,
,
,
3.39
Definition (Maximum and minimum functions.)
We define functions max and min from
to
R by the rule
|
(3.40) |
|
(3.41) |
Thus we have
and
for all
.
Also
3.42
Definition (Absolute value function.)
Let
be defined
by the rule
We call
the
absolute value function and we
usually designate
by
.
3.43
Definition (Sequence)
Let
be a set. A
sequence in
is a function
. I will refer to a sequence in
as a
real sequence.
The sequence is sometimes denoted by . Thus
is the sequence
such
that
for all
. Sometimes the
sequence is denoted by
|
(3.44) |
for example
is the same as
. The notation in formula (3.44) is always
ambiguous. I will use it for sequences like
in which it is somewhat complicated to give an analytic description for
.
If is a sequence, and
, then we often denote by .
3.45
Examples.
Let
denote the set of all polygons in the plane. For each number
in
let
For each
let
and
denote
the polygons inscribed in
and containing
described in section
2.1. Then
and are sequences in .
is a real sequence. (Cf. (2.3) and (2.12).)
is a sequence of intervals.
3.46
Definition (Equality for functions.)
Let
and
be two functions. Then, since a function is
an ordered triple, we have
The rules
and
are equal
if and only if
=
for all
. If
and
then it is customary to write
to mean
.
This is an abuse of notation, but it is a standard practice.
3.47
Examples.
If
is defined by the rule
and
is defined by the rule
then
since
and
have different codomains.
If
and
are defined by the
rules
then
.
In certain applications it is important to know the precise codomain
of a function, but in many applications the precise codomain is not
important, and in such cases I will often omit all mention of the
codomain. For example, I might say ``For each positive number , let
.'' and proceed as though I had defined a function. Here
you could reasonably take the codomain to be
the set of real intervals, or the set of closed intervals, or the set of
all subsets of R.
3.48
Definition (Image of )
Let
be sets, and let
. The set
is called the
image of , and is denoted by
More generally, if
is any subset of
then we define
We call
the
-image of
. Clearly, for every
subset
of
we have
.
3.49
Examples.
If
is defined by the rule
then
.
3.50
Definition (Graph of )
Let
be sets, and let
. The
graph of is defined
to be
If the domain and codomain of are subsets of R, then the graph of
can be identified with a subset of the plane.
3.51
Examples.
Let
be defined by the rule
The graph of
is sketched below. The arrowheads on the graph are intended
to indicate that the complete graph has not been drawn.
Let
Let be the function
from to R defined by the rule
The graph of
is sketched above. The solid dot at
indicates that
is in the graph. The hollow dot at
indicates that
is not in the graph.
Let
be defined by the rule
Thus
and
The graph of
is sketched above.
The term function (functio) was introduced into mathematics
by Leibniz [33, page 272 footnote]. During the
seventeenth century the ideas of function and curve were usually
thought of as being the same, and a curve was often thought of
as the path of a moving point. By the eighteenth century the
idea of function was associated with ``analytic expression''.
Leonard Euler (1707-1783) gave the following definition:
A function of a variable quantity is an analytic expression composed in any way whatsoever of the
variable quantity and numbers or constant quantities.
Hence every analytic expression, in which all component quantities except
the variable are constants,
will be a
function of that ; Thus
; ;
; ; etc.
are functions of [18, page 3].
The use of the notation ``'' to represent the value of
at was introduced by Euler in 1734 [29, page 340].
Next: 3.4 Summation Notation
Up: 3. Propositions and Functions
Previous: 3.2 Sets Defined by
  Index
Ray Mayer
2007-09-07