20
8. September 22nd: Equivalence classes.
8.1
.
Describe explicitly the equivalence classes for the relation
∼
defined in Exer
cise 7.2. How many
distinct
equivalence classes are there? (This gives a description
of the set
Z
/
∼
in this case.)
Answer:
The equivalence relation defined on
Z
in Exercise 7.2 is
a
∼
b
⇐⇒
a
and
b
have the same remainder after division by
n
.
There are
n
possible remainders: 0, 1, . . . ,
n

1. The key observation here is that
if
r
is one of these
n
numbers, then
r
equals the remainder of the division of itself
by
n
.
Thus, no two of these numbers are equivalent.
On the other hand, if
a
has
remainder
r
after division by
n
, then
a
∼
r
, by the same token. Thus, every integer
is equivalent to one of the
n
numbers 0,. . . ,
n

1.
The conclusion is that there are exactly
n
distinct equivalence classes modulo this
equivalence relation. The quotient set
Z
/
∼
is
{
[0]
∼
, . . . ,
[
n

1]
∼
}
.
Here, [
r
]
∼
consists of all integers which have remainder
r
after division by
n
.
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 Fall '11
 Aluffi
 Equivalence relation, equivalence class, equivalence classes, Partition of a set

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