2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS
129
Now for all
x
2
X
,
x
P
x
because there is some
i
2
I
such that
x
2
X
i
. The definition of
x
P
y
is clearly symmetric in
x
and
y
. Finally,
if
x
P
y
and
y
P
z
, then there exist
i
2
I
such that both
x
and
y
are
elements of
X
i
, and furthermore there exists
j
2
I
such that both
y
and
z
are elements of
X
j
. Now
i
is necessarily equal to
j
because
y
is an
element of both
X
i
and of
X
j
and
X
i
\
X
j
D ;
if
i
¤
j
. But then both
x
and
z
are elements of
X
i
, which gives
x
P
z
.
Every partition of a set
X
gives rise to an equivalence relation on
X
.
Suppose that we start with an equivalence relation on a set
X
, form
the partition of
X
into equivalence classes, and then build the equivalence
relation related to this partition. Then we just get back the equivalence
relation we started with! In fact, let
be an equivalence relation on
X
, let
P
D f
OExŁ
W
x
2
X
g
be the corresponding partition of
X
into equivalence
classes, and let
P
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 Fall '08
 EVERAGE
 Algebra, Equivalence relation, Partition of a set

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