This preview shows pages 1–2. Sign up to view the full content.
McCombs Math 381
Working with Functions
1
Given sets
A
and
B
, with a relation
f
from
A
to
B
.
Important Definitions:
1.
f
:
A
→
B
is a
function
from
A
to
B
provided the following are both true:
(i)
( )
b
a
f
B
b
A
a
=
∈
∃
∈
∀
:
,
(ii)
If
() ( )
,
and
c
a
f
b
a
f
=
=
then
b
=
c
.
Given a function
f
:
A
→
B
.
2.
The
domain
of
f
is given by dom
f
=
A
.
T
h
e
codomain
of
f
is given by
codom
f
B
=
.
3.
The
image
of
f
is given by
(
)(
)
{ }
b
a
f
A
a
B
b
A
f
f
=
∈
∃
∈
=
=
with
:
im
.
Note that
image
of
f
is also called the
range
of
f
4.
f
is called a
onetoone
(
injective
) function provided
A
a
a
∈
∀
2
1
,
, if
2
1
a
a
≠
, then
( ) ( )
2
1
a
f
a
f
≠
5.
f
is called an
onto
(
surjective
) function provided
( )
b
a
f
A
a
B
b
=
∈
∃
∈
∀
:
,
In other words,
( )
B
A
f
f
=
=
im
.
6.
f
is called a
bijection
provided
f
onetoone
and
onto.
Important Propositions:
1.
Given
f
:
A
→
B
, the inverse relation
f
−
1
:
B
→
A
is a function
if and only if
f
is a
bijection
.
2.
Given finite sets
A
and
B
, with
a
A
=
and
b
B
=
.
(i)
The total number of functions from
A
to
B
, is given by
b
a
.
(ii)
If
B
A
>
, there are no injections from
A
to
B
.
(iii)
If
B
A
<
, there are no surjections from
A
to
B
.
(iv)
If
B
A
≤
, the total number of injections from
A
to
B
is given by
()
!
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '11
 NA
 Sets

Click to edit the document details