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Equivalence Relations
Aaron Bloomfield
CS 202
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Introduction
•
Certain
combinations
of
relation
properties
are
very
useful
–
We won’t have a chance to see many applications in this course
•
In this set we will study equivalence relations
–
A relation that is reflexive, symmetric and transitive
•
Next slide set we will study partial orderings
–
A relation that is reflexive, antisymmetric, and transitive
•
The
difference
is
whether
the
relation
is
symmetric
or
antisymmetric
3
Equivalence relations
•
A relation on a set
A
is called an
equivalence relation
if it
is reflexive, symmetric, and transitive
•
Consider relation
R
= { (
a,b
) 
len
(
a
) =
len
(
b
) }
–
Where
len
(
a
) means the length of string
a
–
It is reflexive:
len
(
a
) =
len
(
a
)
–
It is symmetric: if
len
(
a
) =
len
(b), then
len
(
b
) =
len
(
a
)
–
It is transitive: if
len
(
a
) =
len
(b) and
len
(b) =
len
(c), then
len
(a) =
len
(c)
–
Thus,
R
is a equivalence relation
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Equivalence relation example
•
Consider the relation
R
= { (
a,b
) 
m

ab
}
–
Called “congruence modulo
m
”
•
Is it reflexive: (a,a)
∈
R
means that
m

aa
–
aa
= 0, which is divisible by
m
•
Is it symmetric: if (
a,b
)
∈
R
then (
b,a
)
∈
R
–
(
a,b
) means that
m

ab
–
Or that
km
=
ab
.
Negating that, we get
ba
= 
km
–
Thus,
m

ba
, so (b,a)
∈
R
•
Is it transitive: if (
a,b
)
∈
R
and (
b,c
)
∈
R
then (
a,c
)
∈
R
–
(
a,b
) means that
m

ab
, or that
km
=
ab
–
(
b,c
) means that
m

bc
, or that
lm
=
bc
–
(
a,c
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This note was uploaded on 11/11/2011 for the course ECON 1110 at Cornell University (Engineering School).
 '06
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