Ma
5
a
HW ASSIGNMENT
1
FALL 09
SOLUTIONS
The exercises are taken from the text,
Abstract Algebra
(third edition) by Dum
mitt and Foote.
We have followed the book’s notation here for
Z
+
as the set of
all positive integers.
(This is not standard, and many use
N
to denote this set,
reserving
Z
+
for the set of nonnegative integers.)
Page 4,
7
.
Proof of Reflexivity:
a
∼
a
⇐⇒
f
(
a
) =
f
(
a
).
Proof of Symmetry:
a
∼
b
⇐⇒
f
(
a
) =
f
(
b
)
⇐⇒
f
(
b
) =
f
(
a
)
⇐⇒
b
∼
a
. Proof of Transitivity:
a
∼
b
and
b
∼
c
⇐⇒
f
(
a
) =
f
(
b
) and
f
(
b
) =
f
(
c
) =
⇒
f
(
a
) =
f
(
c
)
⇐⇒
a
∼
c
.
These all follow from the fact that = is an equivalence relation. The equivalence
class for
a
∈
A
is the set
{
x
∈
A
:
x
∼
a
} ⇐⇒ {
x
∈
A
:
f
(
x
) =
f
(
a
)
}
, which is the
fiber of
f
over
f
(
a
). Therefore the equivalence classes are the fibers of
f
.
Page 8,
6
.
First some philosophy.
If a logical statement
A
=
⇒
B
needs to be
proved, then it is enough to prove a statement equivalent to it, for example its
contrapositive
statement
̸
B
=
⇒ ̸
A
, where
̸
B
(resp.
̸
A
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 Fall '10
 DinakarRamakrishnan
 Algebra, Integers, Natural number, Equivalence relation, Prime number, positive integers

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