2. Let
A
=
{
1
,
2
,
3
}
and
B
=
{
a, b
}
.
Let
f
⊆
A
×
B
be defined by
f
=
{
(1
, a
)
,
(1
, b
)
,
(2
, b
)
,
(3
, a
)
}
. This
f
is not a welldefined function,
since one element of
A
is related to two elements in
B
as is evident
from the diagram:
A
B
1
2
3
a
b
3. The following are examples of functions with infinite domains and co
domains:
(a) the function
f
:
N
×
N
→
N
defined by
f
(
x, y
) =
x
+
y
;
(b) the function
f
:
N
→
N
defined by
f
(
x
) =
x
2
;
(c) the function
f
:
R
→
R
defined by
f
(
x
) =
x
+ 3
.
The binary relation
R
on the reals defined by
x R y
if and only if
x
=
y
2
is not a function, since for example
4
relates to both
2
and

2
.
P
ROPOSITION
4.3 (F
INITE
C
ARDINALITY
)
Let
A
→
B
denote the set of all functions from
A
to
B
, where
A
and
B
are
finite sets. If

A

=
m
and

B

=
n
, then

A
→
B

=
n
m
.
Sketch proof
For each element of
A
, there are
m
independent ways of map
ping it to
B
. You do not need to remember this proof.
4.2
Partial Functions
Recall that it is easy to write Haskell functions which return runtime errors
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 Spring '09
 Koskesh
 Math, Set Theory, Continuous function, Multivalued function, Bijection

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