Types of Relations

A relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in other classes.
1) Reflexive relations- A relation is said to be reflexive provided you take up any element from the given set say a∈A, then the ordered pair (a,a) belongs to that relation i.e for all (a,a) ∈R and this happens for each and every a present in the set.  
Example- A={1,2,3} is the given set and we want the relation to be reflexive then,
(1,1)∈R, (2,2)∈R, (3,3)∈R should be a element in the relation, even if one does not lie then it will not be a reflexive relation. 
(a,a)∈R can also be written as aRa.
2) Symmetric relations- For instance, a,b are the elements from the given set A i.e. a,b∈A and for any (a,b)∈R, if we want this relation to be symmetric then in that case it must imply that (b,a)∈R.
Example- A= {1,2,3} and if R{(1,2), (2,1)} then the relation is symmetric.
3) Transitive relations- For any a,b,c that belongs to set A, if (a,b)∈R and in continuity (b,c)∈R ⇒ (a,c)∈R then we say that the relation is transitive.
Example- A= {1,2,3} and if R= {(1,2), (2,3), (3,2)}
4) Not reflexive- For reflexive relation we write that ∀a∈A ⇒(a,a) ∈R, then for not reflexive it could be said that there exist atleast some element in A such that the ordered pair (a,a) does not belong R, it can be written as ∃a∈A (a,a)∉R, here ∀ means for all and ∃ means there exists. 
5) Not symmetric- For not symmetric there exist some a and b belonging to the given set such that even your( a,b) belongs to R, but (b,a) does not belong to R. This can be expressed as ∃a,b∈A, (a,b)∈R but (b,a) ∉R.
6) Not transitive- There exist three elements in the set such that (a,b), (b,c)∈R, but (a,c) ∉R.
If relation is Reflexive, Symmetric and Transitive all together then the relations is said to be Equivalence Relation.