“Relations: A mathematical approach“

Relation is a word by which we connect at least two quantities by a rule.If there is no connection, it means there is no relation between the quantities, such quantities are treated as independent quantities in mathematics. Generally in algebra we define “a relation is a subset of Cartesian product of two non empty sets”. The definition of a relation indicates that without Cartesian product of two non empty sets a relation cannot be formed. Generally there are three types of relations in algebra Reflexive, symmetric and transitive. The relation which is reflexive and symmetric and transitive is called equivalence relation. Taking this theory in mind the researchers introduced the concept of intuitionistic fuzzy (vague) relations. Fuzzy sets introduced by Zadeh had a great importance in the field of management, computer sciences, and daily life problems. Later on the theory of intutionistic fuzzy set was introduced by Attnassove by using Zadeh’s Fuzzy set theory. In this present paper the author discusses a comparison between relations and vague relations,and their properties.

We are familiar with the theory of crisp sets. A set is well defined collection of objects. If we have two non empty sets A and B, then  a relation is the subset of the Cartesian product  of set A and B. Therefore mathematically suppose R is a relation from A to B, Then R is a set of ordered pairs (a, b) where a  A and b ϵ B.  Every such ordered pair is written as a R b.  If (a, b) do not belongs to R, then a is not related to b. Basically relations can be classified into three categories  Reflexive, symmetric and transitive. The relation which is reflexive and symmetric and transitive is called an equivalence relation. Reflexive relation : If A and B are any two non empty sets, and R be a relation between A and B Then relation R is reflexive iff a R a  a ϵ R. Symmetric relation : If A and B are two non empty sets, and R be a relation between A and B Then relation R is symmetric iff,        a R b → b R a   a, b  R.

Transitive relation: If A and B are any two non empty sets, and R be a relation between A and  B, Then the relation R is Transitive iff.      a R b,  b R c,   → a R  c    a, b, c,     R.

Now, the relation which is reflexive and symmetric and transitive is called an equivalence relation.

When we apply these definitions of relations on daily life we are able to get the real picture of a relation in which we are living. For example if we check the relation “fatherhood”.

By these definitions, then we are able to find exact picture of this relation. For reflexive relation, since no person is the father of himself in this world, therefore the relation of fatherhood is not reflexive. Again if a is the father of b, does not imply that b is the father of a. therefore the relation “fatherhood” is not symmetric, again if a is the father of b, and b is the father of c, does not imply that a is the father of c. therefore the relation “fatherhood” is not reflexive, symmetric, and transitive. Hence this is not  an equivalence relation.

Dr. Hakimuddin Khan

Associate Professor

Dept. of Management Studies