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
are two elements in the same equivalence class, then
f
(
a
)
and
f
(
a
)
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
] =
∅
and
[
a
] = [
b
]
, and
b
∈
[
a
]
and
a
∈
[
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,
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
.
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 Summer '11
 gelfand
 Chemistry, Equivalence relation, Rational number, equivalence class, Congruence relation

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