Types of Relations


  1. One-One Relation(Injective)
  2. Many-one relation
  3. Into relation
  4. Onto relation (Surjective)

Reflexive, symmetric, transitive, not reflexive, not symmetric and not transitive.


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.

If you would like to contribute notes or other learning material, please submit them using the button below. | Relations and Functions part 2 (Empty Relation)

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