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An Introduction to Equivalence Relations and Partitions
Relation
A relation from A to B is a rule that gives us the connection between
elements of A to elements of B. A relation can be between a single set
or it can be between several sets
Formal definition:
Let X and Y be sets. A relation, R, from X to Y is a subset of the
Cartesian product X × Y.
A Cartesian product of X × Y is the super set of any relation between X
and Y. This is because a Cartesian product contain all the possible
relations between X and Y.
∈
The statement (x,y)
R is read "x is related to y by the relation R",
and is denoted by xRy or R(x,y).
Eg.
If X = {1,2,3} and Y = {4,5,6}
Then
R = {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)}
Set X
1
4
2
5
3
6
Set Y
A GRAPHICAL REPRESENTATION OF A RELATION
The total no of possible relations between two sets can be found out as
follows, If X have a number of elements and Y have b number of
elements, Then total number of possible relations between set X and Y
will be 2^ab.
Properties of Relations:
• Reflexive
• Symmetric
• Transitive
•
Reflexive
A relation in which all the elements follow the property AA.
i.e. all the elements are related to themselves is known as
reflexive relation.
If for all x in set X, the relation G = xRx holds true then G is
said to be reflexive.
•
Symmetric
A relation in which all the elements follow the property such
that, if AB then BA is said to be Symmetric relation.
If for all x and y in X, the relation G = xRy = yRx holds true
then G is said to be Symmetric.
•
Transitive
A relation in which all the elements follow the property such
that, if AB and BC then AC is said to be Transitive
relation.
It is possible that a relation may not have any one of the above
mentioned properties, it may have some of these properties or It might
agree to all the properties.
Binary relations
A binary relation is an ordered pair between the elements of two sets.
It is a subset of the Cartesian product of those two sets.
A binary relation R can be defined as an ordered triple (X, Y, G) where
X and Y are sets, and G is a subset of the Cartesian product X × Y.
∈
The statement (x,y)
R is read "x is related to y by the relation R",
and is denoted by xRy or R(x,y).
Eg.
If X = {1,2,3,4,5,6} and Y = {1,4,16,25,36}
Then
X × Y = {(1,1), (2,4), (4,16), (5,25), (6,36)}
Set X
1
4
16
25
36
1
2
3
4
5
6
Set Y
A GRAPHICAL REPRESENTATION OF A BINARY RELATION
Inverse of a Binary Relation:
An Inverse of a binary relation R is denoted by R
as R -1 = {(y, x) | (x, y)
R }.
∈
-1
and is defined
Now an inverse of a relation can be equal to the relation
i.e. R -1 = R. For this to happen the condition required is the
relation R should be symmetrical.
Equivalence Relation:
An equivalence relation is denoted by “~”
A relation is said to be an equivalence relation if it adheres to the
following three properties mentioned in the earlier part of this paper.
i.e. For a relation to known as equivalence it should fulfill all the
following properties:
•
•
•
Reflexive
Symmetric
Transitive
For all x, y, z in X
1. if x~x
2. if x~y then y~x
3. if x~y and y~z then x~z
Let X be a set and let x, y, and z be elements of X. An equivalence
relation, ~, on X is a relation on X such that:
Reflexive Property: x is equivalent to x for all x in X.
Symmetric Property: if x is equivalent to y, then y is equivalent to x.
Transitive Property: if x is equivalent to y and y is equivalent to z,
then x is equivalent to z.
Examples of equivalence relations:
• equality (=) relation between elements of any set.
Suppose A, B and C are three equal sets of natural numbers. Let
a, b, c be the elements of the set A, B, C.
Reflexive Property: a is equivalent to a for all a in A.
Symmetric Property: if a is equivalent to b, then b is equivalent
to a.
Transitive Property: if a is equivalent to b and b is equivalent
to c, then a is equivalent to c.
•
"similar to" or "congruent to" on the set of all triangles.
Reflexive Property: Triangle A is similar to triangle A
Symmetric Property: If A is similar to B then B is also similar
to A.
Transitive Property:If A is similar to B and B is similar to C
then A is similar to C.
•
is parallel to in case of lines in space.
Reflexive Property :Line A is similar to line A
Symmetric Property :If A is similar to B then B is also similar
To A.
Transitive Property:If A is similar to B and B is similar to C
then A is similar to C.
Examples of relations that are not equivalences:
In the below given examples one of the three mandatory properties of
the equivalence fails and hence these relation are non equivalence.
•
The relation "≥" between real numbers is reflexive and
transitive, but not symmetric. For example, 7 ≥ 5 does not imply
that 5 ≥ 7.
Reflexive Property: 5 ≥ 7
Symmetric Property: if 5 ≥ 7 then the other way round is not
possible.
– False Relation.
Transitive Property: 5 ≥ 7 and 7 ≥ 9 then 5 ≥ 9.
•
is a sibling of
Reflexive Property: A is not a sibling of A.
-- False Relation.
Symmetric Property: A is a sibling of B and B is a sibling of A.
Transitive Property: A is a sibling of B and B is a sibling of C
then A is a sibling of C if A ≠ C
Partition:
A partition of a set X is a set of nonempty subsets of X such that
every element x in X is in exactly one of these subsets.
A set P of subsets of X, is a partition of X if
1. No element of P is empty.
2. The union of the elements of P is equal to X.
3. The intersection of any two elements of P is empty.
Examples
For any non-empty proper subset A of a set U, this A together with its
complement is a partition of U.
The set { 1, 2, 3 } has these five partitions.
o
o
o
o
o
{
{
{
{
{
{1}, {2}, {3} }
{1, 2}, {3} }
{1, 3}, {2} }
{1}, {2, 3} }
{1, 2, 3} }
Note that
o { {}, {1,3}, {2} } is not a partition.
o { {1,2}, {2, 3} } is not a partition.
o { {1}, {2} } is not a partition.
If an equivalence relation is given on the set X, then the set of all
equivalence classes forms a partition of X. Conversely, if a partition
P is given on X, we can define an equivalence relation on X by writing
x ~ y if there exists a member of P which contains both x and y. The
notions of "equivalence relation" and "partition" are essentially
equivalent.
Reference:
1. http://www.wikipedia.org/
2. http://www.iscid.org/encyclopedia/Binary_Relation
3. http://www.math.csusb.edu/notes/rel/rel.html