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Equivalence Relations

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					                           Equivalence Relations
Equivalence Relations

A relation on A is an equivalence relation if it is reflexive, symmetric and transitive. An
example of such is equality on a set. One might think of equivalence as a way to glob together
elements that can be considered the same relative to a property. That is elements become
indistinguishable relative to the relation. For example in arithmetic we don't think twice about
1/2 and 2/4 as having the same value but they are different objects. In geometry, similarity of
triangles is an equivalence relation. A right angled triangle with legs of length 3 and 4 and
hypotenuse of length 5 is not the same as one with lengths 6, 8 and 10. Yet, we think of them
as equivalent because the ratios of the corresponding sides are the same.

For each , we define the equivalency class containing a to be the set of those elements
which are equivalent to a. We denote this set by [a] although you should be aware that there
is no standard notation for this. Some authors use , <a> and so on. The equivalence classes
have some interesting properties. First, no equivalence class is empty since                . Two
equivalence classes are either equal or disjoint. In fact, a and b are equivalent if and only if [a]
= [b]. Each element of A belongs to some equivalence class ( in fact ). The union of all the
equivalence classes is A. We say that the equivalence classes partition A. A partition of A is a
collection of non empty disjoint subsets of A and whose union is A. Thus the equivalence
classes of a relation are a partition.
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Each equivalence relation on a set partitions the set into its equivalence classes but also for
each partition of the set there is an equivalence relation whose equivalence classes are the
sets in the partition. As an example, if      one partition is    . Consequently, there is an
equivalence relation on A whose equivalence classes are these two subsets.

One problem is how should we refer to the equivalence classes? As we noted above [a] = [b]
whenever a and b are equivalent. What we do is choose one and only one element of each
equivalence class to represent the class. That element is called a class representative. The
set of representatives is called the set of equivalence class representatives. There are usually
many such ways to construct this set of representatives.

If is an equivalence relation on A then the factor set denoted by is the set consisting of the
equivalence classes of . For the example we gave above, our equivalence classes are [1]
and [3]. Thus the factor set is {[1], [3]}. We have chosen 1 to represent the equivalence class
{1,2} and (no choice) 3 to represent the equivalence class {3}. Thus the set of class
representatives is {1,3}. Note the similarity with the factor set. The only difference is that the
elements of the factor set have [ and ] around the number (and of course are subsets of A
whereas the things in the set of class representatives are elements of A). Because of this, we
often abuse notation and write the set of class representatives for the factor set, letting the
context make it clear how we are using a class representative.

An equivalence relation on a set is a subset of , i.e., a collection of ordered pairs of elements
of , satisfying certain properties. Write "" to mean is an element of , and we say " is related
to ," then the properties are

1. Reflexive: for all ,
2. Symmetric: implies for all
3. Transitive: and imply for all ,

where these three properties are completely independent. Other notations are often used to
indicate a relation, e.g., or .
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In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so
that every element of the set is a member of one and only one cell of the partition. Two
elements of the set are considered equivalent (with respect to the equivalence relation) if and
only if they are elements of the same cell. The intersection of any two different cells is empty;
the union of all the cells equals the original set.

Although various notations are used throughout the literature to denote that two elements a
and b of a set are equivalent with respect to an equivalence relation R, the most common are
"a ~ b" and "a ≡ b", which are used when R is the obvious relation being referenced, and
variations of "a ~R b", "a ≡R b", or "aRb".

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is
reflexive, symmetric and transitive. Equivalently, for all a, b and c in A:

a ~ a. (Reflexivity)

if a ~ b then b ~ a. (Symmetry)

if a ~ b and b ~ c then a ~ c. (Transitivity)

A together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted
[a], is defined as .

Reflexivity follows from symmetry and transitivity if for every element a ∈A, there exists
another element b∈A such that a~b holds. However, reflexivity does not follow from symmetry
and transitivity alone. For example, let A be the set of integers, and let two elements of A be
related if they are both even numbers. This relation is clearly symmetric and transitive, but in
view of the existence of odd numbers, it is not reflexive.

On the other hand, let A be the set of integers, and let two elements of A be related if their
difference is even. This is an equivalence relation, which partitions the integers into two
equivalence classes, the even and odd integers.



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posted:7/23/2012
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