Section 7.5 Equivalence Relations

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Section 7.5Equivalence Relations

• Longin Jan Latecki
• Temple University, Philadelphia
• latecki_at_temple.edu

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We can group properties of relations together to
define new types of important relations. ________
_________ Definition A relation R on a set A is
an equivalence relation iff R is reflexive
symmetric transitive Two elements related by
an equivalence relation are called
equivalent. Examples of equivalence relations
Ex. 1, p. 508 Ex. 4, p. 509
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An equivalence class of an element x x y
ltx, ygt is in R x is the subset of all
elements related to x by R. The element in the
bracket is called a representative of the
equivalence class. We could have chosen any
one. Theorem Let R be an equivalence relation
on A. Then either a b or a nb
F The number of equivalence classes is called
the rank of the equivalence relation.
Let Aa,b,c and R be given by a digraph
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Theorem Let R be an equivalence relation on a
set A. The equivalence classes of R partition the
set A into disjoint nonempty subsets whose union
is the entire set. This partition is denoted A/R
and called the quotient set, or the partition
of A induced by R, or, A modulo R. Definition
Let S1, S2, . . ., Sn be a collection of subsets
of a set A. Then the collection forms a partition
of A if the subsets are nonempty, disjoint and
exhaust A
Note that , 1,3, 2 is not a partition
(it contains the empty set). 1,2, 2, 3
is not a partition because . 1, 2 is
not a partition of 1, 2, 3 because none of its
blocks contains 3.
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It is easy to recognize equivalence relations
using digraphs The equivalence class of a
particular element forms a universal relation
(contains all possible arcs) between the elements
in the equivalence class. The (sub)digraph
representing the subset is called a complete
(sub)digraph, since all arcs are
present. Example All possible equivalence
relations on a set A with 3 elements
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1. Determine whether the relations represented by
these zero-one matricesare equivalence
relations. If yes, with how many equivalence
classes?
2. What are the equivalence classes (sets in the
partition) of the integersarising from
congruence modulo 4? 3. Can you count the number
of equivalence relations on a set A with n
elements. Can you find a recurrence relation? The
answers are 1 for n 1 2 for n 2 5 for n
3 How many for n 4?
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Theorem (Bell number) Let p(n) denotes the number
of different equivalence relations on a set with
n elements (which is equivalent to the number of
partitions of the set with n elements). Then
p(n) is called Bell number, named in honor of
Eric Temple Bell Examples p(0)1, since there
is only one partition of the empty set into the
empty collection of subsets p(1)C(0,0)p(0)1,
since 1 is the only partition of
1 p(2)C(1,0)p(1)C(1,1)p(0)115, since
portions of 1,2 are 1,2 and
1,2 p(3)5, since, the set 1, 2, 3 has
these five partitions. 1, 2, 3 ,
sometimes denoted by 1/2/3. 1, 2, 3 ,
sometimes denoted by 12/3. 1, 3, 2 ,
sometimes denoted by 13/2. 1, 2, 3 ,
sometimes denoted by 1/23. 1, 2, 3 ,
sometimes denoted by 123.
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Proof (Bell number)
We want to portion 1, 2, , n. For a fixed j, A
is a subset of j elements from 1, 2, , n-1
union n. Note that j can have values from 0 to
n-1. We can select a subset of j elements from
1, 2, , n-1 in C(n-1,j) ways, and we have
p(n-1-j) partitions of the remaining n-1-j
elements.
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Theorem If R1 and R2 are equivalence relations
on A, then R1nR2 is an equivalence relation on
A. Proof It suffices to show that the
intersection of reflexive relations is
reflexive, symmetric relations is symmetric,
and transitive relations is transitive.
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Definition Let R be a relation on A. Then the
reflexive, symmetric, transitive closure of R,
tsr(R), is an equivalence relation on A, called
the equivalence relation induced by R. Example
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Theorem tsr(R) is an equivalence
relation. Proof We need to show that tsr(R) is
still symmetric and reflexive. Since we only
add arcs vs. deleting arcs when computing
closures it must be that tsr(R) is reflexive
since all loops ltx, xgt on the diagraph must be
present when constructing r(R). If there is an
arc ltx, ygt then the symmetric closure of r(R)
ensures there is an arc lty, xgt. Now argue that
if we construct the transitive closure of sr(R)
and we add an edge ltx, zgt because there is a path
from x to z, then there must also exist a path
from z to x (why?) and hence we also must add an
edge ltz, xgt. Hence the transitive closure of
sr(R) is symmetric.