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Mathematics 113: Homework # 1
Curtis Tekell
Dr. Poirier
September 8, 2010
Exercise 1.
Determine whether the following functions
f
are well deﬁned:
(a)
f
:
Q
→
Z
deﬁned by
a
b
7→
a
.
(b)
f
:
Q
→
Q
deﬁned by
a
b
7→
a
2
b
2
.
Solutions.
(a) No. 1 =
f
(
1
2
) =
f
(
2
4
) = 2, hopefully not true.
(b) Yes, this is well deﬁned. We do not have a problem with choosing arbitrary
representatives from the equivalence classes
a
b
. For instance, assume that
a
b
=
c
d
.
Then
ad
=
bc
, and
a
2
b
2
=
f
(
a
b
) =
f
(
c
d
) =
c
2
d
2
.
Therefore
a
2
d
2
=
c
2
b
2
, which holds if
ad
=
bc
.
Exercise 2.
Let
A
be a nonempty set.
(a) If
∼
determines an equivalence relation on
A
,
A/
∼
partitions
A
.
(b) If
D
=
{
A
i

i
∈
I
}
is a partition of
A
with no
A
i
=
∅
, then there exists an
equivalence relation
∼
on
A
such that
A/
∼
=
D
.
Proof.
(a) First, as
∼
is an equivalence relation, it is reﬂexive, thus
a
∼
a
for all
a
∈
A
,
so each
a
is in some equivalence class, and
S
D
∈
A/
∼
D
=
A
, it remains to show
that no nonequal
B,C
∈
A/
∼
intersect. Assume that
a
∈
B
∩
C
. Clearly
a
∼
c
and
a
∼
b
for all
b
∈
B
and
c
∈
C
, thus by transitivity,
b
∼
c
for all
b
∈
B
and
c
∈
C
, and hence
C
=
B
. It follows that two equivalence classes are either
equal or disjoint, and thus
A/
∼
deﬁnes a partition on
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 Summer '11
 gelfand
 Chemistry

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