2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS
127
Definition 2.6.2.
A
partition
of a set
X
is a collection of mutually disjoint
nonempty subsets whose union is
X
.
Equivalence relations and set partitions are very common in mathe
matics. We will soon see that equivalence relations and set partitions are
two aspects of one phenomenon.
Example 2.6.3.
(a)
For any set
X
, equality is an equivalence relation on
X
. Two
elements
x; y
2
X
are related if, and only if,
x
D
y
.
(b)
For any set
X
, declare
x
y
for all
x; y
2
X
.
This is an
equivalence relation on
X
.
(c)
Let
n
be a natural number.
Recall the relation of
congruence
modulo
n
defined on the set of integers by
a
b .
mod
n/
if, and
only if,
a
b
is divisible by
n
. It was shown in Proposition
1.7.2
that congruence modulo
n
is an equivalence relation on the set
of integers. In fact, this is a special case of the coset equivalence
relation, with the group
Z
, and the subgroup
n
Z
D f
nd
W
d
2
Z
g
.
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 Fall '08
 EVERAGE
 Algebra, Sets, Equivalence relation, Euclidean geometry, equivalence class

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