COT3100.01, Fall 2000
Assigned: 10/19/2000
S. Lang
Assignment #4 (40 pts.)
Due: 10/31 in class (within
the first 10 minutes)
First, we
define
when two functions are considered equal (or identical).
Definition.
Two functions
f
:
A
→
B
and
g
:
A
→
B
are said
equal
, denoted
f
=
g
, if
f
(
x
) =
g
(
x
) for
every
x
∈
A
, where
A
is the common domain of the two functions.
For example, two functions
f
(
x
) = (
x
+ 1)
2
and
g
(
x
) =
x
2
+ 2
x
+ 1, both defined from
R
to
R
, where
R
denotes the set of real numbers, are equal because (
x
+ 1)
2
=
x
2
+ 2
x
+ 1 by algebra laws.
As a reminder, when there are two functions
f
:
A
→
B
and
g
:
B
→
C
, they can be composed to
form a function denoted
g
o
f
:
A
→
C
, with
g
precedes
f
in the composition notation
g
o
f
.
However, when
f
and
g
are considered as relations (because functions are special cases of
relations), the notation for composing would be
f
o
g
, which as a relation has the property
f
o
g
⊂
A
×
C
.
The context should make it clear which convention (either
g
o
f
or
f
o
g
) is used.
1.
(11 pts.) Consider a set
A
= {
a
,
b
,
c
}, a set
B
= {1, 2}, and a set
C
= {
u
,
v
,
w
}.
Define a
relation
R
⊂
A
×
B
with
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 Fall '09
 Set Theory, Inverse function, Bijection

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