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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 2 3 4 5 6 1 4 16 25 36 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 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 Symmetric Property : a is equivalent to a for all a in A. : 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 Symmetric Property : Triangle A is similar to triangle A : 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 Symmetric Property :Line A is similar to :If A is similar to B to A. Transitive Property :If A is similar to B then A is similar to line A then B is also similar and B is similar to C 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 Symmetric Property : 5 ≥ 7 : 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 Symmetric Property : A is not a sibling of A. -- False Relation. : 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