Introduction to Abstract Algebra
Homework 1
September 8, 2010
1 Section 0.1, Excercise 5:
Determine whether the following functions are well deﬁned:
(a)
f
:
Q
→
Z
deﬁned by
f
(
a
b
)
=
a
.
Our criteria for a well deﬁned function is, that all representatives of the same equivalence
class map to elements in the same equivalence class.
For example, if
a
and
a
Í
are two elements in the same equivalence class, then
f
(
a
) and
f
(
a
Í
) must be in the same equivalence class for the function to be well deﬁned.
This is not the case for this function. For example,
2
2
and
1
1
are in the same equivalence
class, but map to elements in diﬀerent equivalence classes.
(b)
f
:
Q
→
Q
deﬁned by
f
(
a
b
)
=
a
2
b
2
.
This function is well deﬁned, since all numbers in the same equivalence class map to
elements in the same equivalence class, e.g. 3/4 and 6/8 map to 9/16 and 36/64 respectively.
These are equivalent, since 36/64 = 9/16.
2 Prove Proposition 2, section 0.1:
Let
A
be a nonempty set. We wish to prove that
(a)
Theorem 1.
If
∼
determines an equivalence relation on
A
, then the set of equivalence
classes of
A/
∼
of
∼
forms a partition of
A
.
Proof.
The proof consists of two parts. We ﬁrst prove, that any two equivalence classes
are distinct. Second we prove, that the union of all equivalence classes make up all of
A
.
(a) Let
a,b
∈
A
. Assume that [
a
]
∩
[
b
]
Ó
=
∅
and [
a
]
Ó
= [
b
], and
b
Ó∈
[
a
] and
a
Ó∈
[
b
]. Now
suppose there exists some
x
∈
[
a
] and
x
∈
[
b
]. Then
x
∼
a
and
x
∼
b
. Due to
symmetry,
a
∼
x
. Now, from the property of transitivity,
a
∼
b
, which means
a
∈
[
b
],
so we have a contradiction, and the sets must be equal or distinct.
(b) For all
a
∈
A
:
a
∈
[
a
], so clearly the union of all such [
a
] cover all of
A
. Formally put,
Û
i
∈
I
A
i
=
A.
(b)
Theorem 2.
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
.
Proof.