Relation

 Relation:

                A relation R over a set X can be seen as a set of ordered pairs (x, y) of members of X.

               X= {1,2,3,4,5,6,7}

               the relation "is less than" on the set X is given by R= {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)}

                If we consider social relationships like fatherhood, brotherhood, sonhood, etc. 

                For example, Ram is the son of Dashrath, Sita was the daughter of Janak, etc.

 

Types of Relations:

• Empty Relation: 

                    A relation R on a set A is called Empty if the set A is empty set.

• Reflexive Relation:

                 A relation R on a set A is called reflexive if (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

• Irreflexive relation:

                A relation R on a set A is called irreflexive if no (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, b), (b, a)} is irreflexive relation. 

 

• Symmetric Relation:

                A relation R on a set A is called symmetric if (b,a) € R holds when (a,b) € R.i.e. The relation R= {(4,5), (5,4), (6,5), (5,6)} on set A= {4,5,6} is symmetric.

 

• Transitive Relation:

                A relation R on a set A is called transitive if (a,b) € R and (b,c) € R then (a,c) € R for all a,b,c € A.i.e. Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.

 

• Asymmetric relation:

                Asymmetric relation is opposite of symmetric relation. A relation R on a set A is called asymmetric if no (b,a) € R when (a,b) € R.