Types of Relation

What are the Types of Relations?

1. Empty Relation

2. Universal Relation

3. Reflexive Relation

4. Symmetric Relation

5. Transitive Relation

1   1.     What is Empty Relation?

                       A Relation R in any set say A is said to be Empty Relation if no elements of                               Set A is related no elements of itself.

                              i.e.  R = Ф ⊃ A×A.

2.      What is Universal Relation? 

                                  A Relation R in any set say A is said to be Universal Relation if all elements of Set A is related to its all elements.

                            i.e.  R = A×A.

     3.  What is Reflexive Relation? 

                             A Relation R in any set say A is said to be Reflexive Relation if all same elements present in any Relation.

     i.e. if (a, a) ∊ R, for every a ∈ A.

     For Example R = {(1, 1), (2, 2), (a, a), (x, x)}

    

         4.   What is Symmetric Relation?

                            A Relation R in any set say A is said to be Symmetric Relation if all same elements present in any Relation R then its mirror image also presents in that Relation.

      i.e. if (a, b) ∊ R,  then it implies (b, a) ∊ R for all a, b ∈ A.

      For Example R = {(1, 2), (2, 1), (a, b), (b, a), (x, y), (y, x)}

    

     5. What is Transitive Relation?  

                              A Relation R in any set say A is said to be Transitive Relation 

       i.e. if (a, b) ∊ R,  (b, c) ∊ R then it implies (a, c) ∊ R for all a, b, c ∈ A.

       For Example R = {(1, 2), (2, 1), (1, 1), (b, a), (a, y), (b, y)}

      

        Note:    Equivalence Relation:

                        A Relation R which is Reflexive, Symmetric and Transitive said to be equivalence Relation.