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Introduction to Ring Theory. Math 228
Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra,
Second Edition.
Homework 3  due February 2, 2010
D.3(a): Prove that the following relation on the set
R
of real numbers is an equivalence
relation:
a
∼
b
if and only if cos
a
= cos
b
.
Solution:
It is reﬂexive: cos
a
= cos
a
, therefore
a
∼
a
.
It is symmetric:
a
∼
b
iﬀ cos
a
= cos
b
, iﬀ cos
b
= cos
a
iﬀ
b
∼
a
.
It is transitive:
a
∼
b
and
b
∼
c
, iﬀ cos
a
= cos
b
and cos
b
= cos
c
. But that implies
cos
a
= cos
c
, which implies
a
∼
c
.
A relationship that is reﬂexive, symmetric, and transitive is an equivalence relation.
2.1.12: Which of the following congruences have solutions:
(a)
x
2
≡
1 (mod 3)
(b)
x
2
≡
2 (mod 7)
(c)
x
2
≡
3 (mod 11)
Solution:
(a) Since
x
≡
0 (mod 3)
⇒
x
2
≡
0 (mod 3)
x
≡
1 (mod 3)
⇒
x
2
≡
1 (mod 3)
x
≡
2 (mod 3)
⇒
x
2
≡
1 (mod 3)
then
x
2
≡
1 (mod 3) has a solution for
x
≡
1 or 2 (mod 3)
(b) Since
x
≡
0 (mod 7)
⇒
x
2
≡
0 (mod 7)
x
≡
1 (mod 7)
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 Fall '10
 GIESEKER,D.
 Algebra, Real Numbers

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