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40
1. ALGEBRAIC THEMES
We denote by
Z
n
the set of residue classes modulo
n
. The set
Z
n
has a natural algebraic structure which we now describe. Let
A
and
B
be
elements of
Z
n
, and let
a
2
A
and
b
2
B
; we say that
a
is a
representative
of the residue class
A
, and
b
a representative of the residue class
B
.
The class
Œa
C
bŁ
and the class
ŒabŁ
are independent of the choice of
representatives. For if
a
0
is another representative of
A
and
b
0
another
representative of
B
, then
a
±
a
0
.
mod
n/
and
b
±
b
0
.
mod
n/
; therefore
a
C
b
±
a
0
C
b
0
.
mod
n/
and
ab
±
a
0
b
0
.
mod
n/
according to Lemma
1.7.5
. Thus
Œa
C
bŁ
D
Œa
0
C
b
0
Ł
and
ŒabŁ
D
Œa
0
b
0
Ł
. This means that it makes
sense to deﬁne
A
C
B
D
Œa
C
bŁ
and
AB
D
ŒabŁ
. Another way to write
these deﬁnitions is
ŒaŁ
C
ŒbŁ
D
Œa
C
bŁ;
ŒaŁŒbŁ
D
ŒabŁ:
(1.7.1)
Example 1.7.6.
Let us look at another example in which we
cannot
deﬁne
operations on classes of numbers in the same way (in order to see what
the issue is in the preceding discussion). Let
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 Fall '08
 EVERAGE
 Algebra

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