what is Relation, reflexive relation, symmetric relation, asymmetric relation, transitive relation, equivalence relation, partial order relation

 The relation is an association between two or more things. relation between sets is the association between then that defines how the sets are related.

 Types of relations:

Reflexive relation

If a set A have elements like {a,b,c,d} the relation, must contain all (a,a), (b,b),(c,c),(d,d) in it then the relation is called as reflexive relation. in simple words the relation which contain (a,a) pair for all a n=belongs to the set, then the relation  on the set is called as reflexive relation. each element present in the set must be related to it self then the relation is called as reflexive relation.
for example: A={1,2,3,6,8} the relation on set A is such that it defines (a,b) belongs to A when b is divisible by a.

here R={(1,1),(1,2),(1,3),(1,6),(1,8),(2,2),(2,6),(2,8),(3,3),(3,6),(6,6),(8,8)} 

here in the above relation all pairs that are like (a,a) are present in relation R that are: (1,1),(2,2),(3,3),(6,6),(8,8) all are present in the relation set so the above relation on set A is reflexive relation.

Symmetric relation

If (a,b) belongs to the relation then (b, a) must belong to the same relation then the relation is called symmetric relation.  in simple words when a is related to b  i.e (a,b) then b should be related with a is (b,a) should be present in the relation set. the elements which are related to other element the other element must be related to the element then the relation is called as symmetric relation.

for example:

A={1,2,3,4,5} the relation R on set A  is {(1,1),(1,2),(2,1),(3,3),(4,5),(5,4)}

to check whether the given relation is a symmetric relation or not,  we should check that each pair in the relation that is (a,b) there must must present (b,a). for pair (1,1) the symmetric pair will be the same. for pair (1,2) there must present (2,1) and the pair (2,1) is present in the relation. for next pair (3,3) the symmetric pair will be the same. for the next pair (4,5) symmetric pair is (5,4) and the pair is present in the relation set. so the given relation is a symmetric relation.

Transitive relation

If (a,b) belongs to the relation and (b,c) also belongs to the relation then there must exist the (a,c) then the relation is called the transitive relation.

for example:

A={1,2,3} the relation on set A is R ={(1,1),(1,2),(2,1),(2,2),(2,3),(3,3)}

to check whether the given relation is transitive or not  we must check the transitive property for each pair in the relation R on set A. for the first pair (1,1) and (1,2) belongs to the relation set and the (1,2) that is its transitive pair is also present in it. for the second pair (1,2) (2,1) are present then (1,1) must be present and the pair is present in the relation set. for the third pair (2,1) and (1,2) are present then (2,2) must be present and the pair is present in the relation set. for the fourth pair (2,2) and (2,1) are present then (2,2) must be present and the pair is present in the relation set. for the fifth pair (2,3) and (3,3) are present then (3,3) must be present and the pair is present in the relation set. for the last pair (3,3) is is transitive because for the pair (3,3) and (3,3) the transitive pair will be (3,3) so the given relation is the transitive relation.

Asymmetric relation

If (a,b) belongs to the relation then (b, a)  not belong to the same relation then the relation is called asymmetric relation, in simple words when a is related to b  i.e. (a,b) then b should not be related with a is (b,a) should not present in the relation set. the relation which is symmetric is not asymmetric relation.

for example:

A={1,2,3} the relation on set A is R={(1,1),(2,1),(1,3)(2,3)} 

here to check whether the given relation is asymmetric relation or not we should first check for the asymmetric property for the given relation. for checking for first pair (1,1) which is not like (a,b) so there is no need to check for (b,a). for second pair(2,1) there is no pair present which is (1,2) so this pair is showing asymmetric relation. for third pair  (1,3) there is no pair present (3,1) so this pair is showing asymmetric property. for fourth pair (2,3) . there is not any pair (3,2) so this pair is showing asymmetric property. so the given relation is an asymmetric relation.

Antisymmetric relation 

If  (a,b) belongs to the relation set then (b,a) must not be present in the relation set, if both (a,b) and (b,a) present the a must be equal to b.

for example:
A={1,2,3,4} then R on set A is R={(1,1),(2,2),(3,3)} by checking the property of antisymmetricity we can get that the given relation is antisymmetric relation.

Equivalence relation

The relation which is reflexive, symmetric and transitive is called an equivalence relation.  Equivalence relation must follow all the properties of three relations (reflexive, symmetric and transitive).

for example:

A={a,b,c} the relation R={(a,a),(b.b),(b,c),(c,c),(c,b)}

to check whether the given relation is equivalence relation or not. we must check the given relation is reflexive or not, in the given relation every (a,a) pair is present so the given relation is a reflexive relation. Now check for symmetric property given relation contains  (b,c) and (c,b) and hence the given relation is symmetric relation. finally check for transitive property, given relation contains transitive pairs and hence the given relation is equivalence relation.

Partial order relation

A relation which is  reflexive, antisymmetric and transitive is called a partial order relation. Partial order relation must follow the properties of the three relations(reflexive, antisymmetric and transitive).