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
.
The
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
(
)
!
!
b
b
a
−
.
(v)
If
B
A
=
, the total number of bijections from
A
to
B
is given by
!
a
.
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 Summer '11
 NA
 Sets, Inverse function, Finite set, McCombs Math

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