Math 113 Section 5
Homework #1
Fall 2010
Due: Wednesday, September 8
1.
Section 0.1, Exercise 5:
Determine whether the following functions
f
are well defined:
(a)
f
:
Q
→
Z
defined by
f
(
a
b
) =
a
.
(b)
f
:
Q
→
Q
defined by
f
(
a
b
) =
a
2
b
2
.
2.
Prove Proposition 2, section 0.1:
Let
A
be a nonempty set.
(a)
Prove that if
∼
determines an equivalence relation on
A
then the set of equivalence
classes
A/
∼
of
∼
forms a partition of
A
.
(b)
Prove that if
{
A
i

i
∈
I
}
is a partition of
A
then there is an equivalence relation
on
A
whose equivalence classes are precisely the sets
A
i
, i
∈
I
.
3.
Let
Z
>
0
=
{
1
,
2
, . . .
}
be the set of positive integers (sometimes called the natural
numbers and denoted
N
). Strange but true: there is a bijection of sets
ϕ
:
Q
→
Z
>
0
.
Determine
ϕ
.
Hint: you may describe
ϕ
as a list of ordered pairs or as a formula or as any kind of
diagram you like. The definition need not be unique so your answer may differ from
your friend’s.
4.
(a)
Use the Euclidean algorithm to determine the greatest common divisor of
12
,
345
and
9876
.
(b)
Find
a
and
b
so that
12
,
345
a
+ 9
,
876
b
= (12
,
345
,
9
,
876)
.
5.
Section 0.2, Exercise 7:
If
p
is prime, prove that there do not exist
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 Fall '08
 OGUS
 Math, Algebra, Prove Proposition, Z/12Z

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