Types of Relation in Mathematics
Types of relation
Void relation:-
Let A be a set. Then, Φ ⊆ A x A and so it is a relation on A. This relation is called the void or empty relation on set A.
In other words, a relation R on a set A is called void or empty relation, if no element of A is related to any element of A.
Consider the relation R on the set A = {1, 2, 3, 4, 5} defined by R {(a, b) : a – b = 12} . We observe that a – b ≠ 12 for any two elements of A
(a, b) ∉ R for any a, b ∈ A.
⇒ R does not contain any element of A x A
⇒ R is empty set
⇒ R is the void relation on A.
Universal relation:-
Let A be a set. Then, A x A ⊆ A x A and so it is a relation on A.This relation is called the universal reIation on A.
In other words, a relation R on a set is called universal relation,if each element of A is related to every element of A.
Consider the relation R on the set A ={1, 2, 3, 4, 5, 6} defined by R = {(a, b) ∈ R 😐 a – b | ≥ 0} . We observe that
|a – b| ≥ 0 for all a, b ∈ A
⇒ (a,b) ∈ R for all (a,b) ∈ A x A
⇒ Each element of set A is related to every element of set A
⇒ R = A x A
⇒ R is universal relation on set A
NOTE :- It is to note here that the void relation and the universal relation on a set A are respectively the smallest and the largest relations on set A. Both the empty (or void) relation and the universal relation are sometimes called trivial relations.
Identity of relation:-
Let A be set. Then the relation IA= {(a, a) : a ∈ A} on A is called the identity relation on set A.
In other words, a relation IA on A is called the identity relation if every element of A is related to itself only.
If A = {1, 2, 3} , then the relation IA = {(1,1), (2, 2), (3, 3)} is the identity relation on set A. But, relations R1= {(1, 1), (2, 2)} and R2 ={(1,1), (2, 2), (3, 3), (1, 3)} are not identity relations on A, because ( 3, 3) ∉ R1 and in R2 element 1 is related to elements 1 and 3.
Reflexive relation:-
A relation R on a set A is said to be reflexive if every element of A is related to itself.
Thus, R is reflexive ⇔ (a, a) ∈ R for all a ∈ A.
A relation R on a set A is not reflexive if there exists an element a ∈ A such that (a, a) ∉ R.
Example
Example
Let A = {1, 2, 3} be a set. Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A. But, R1= {(1, 1), (3, 3), (2, 1), (3, 2)} is not a reflexive relation on A, because 2 ∈ A but (2, 2) ∉ R1.
Symmetric relation:-
A relation R on a set A is said to be a symmetric relation iff(if and only if)
(a, b) ∈ R⇒(b, a) ∈ R for all a, b ∈ R
i.e. aRb ⇒bRa for all a, b ∈ A.
Example
-
The identity and the universal relations on a non-void set are symmetric relations.
Transitive relation:-
Let A be any set. A relation R on A is said to be a transitive relation iff(if and only if) (a, b) ∈ R and (b, c) ∈ R ⇒(a, c) ∈ for all a, b, c ∈ R
i.e. aRb and bRc ⇒ aRc for all a, b, c ∈ A.
Example
-
The identity and the universal relations on a non-void set are transitive relations.
Antisymmetric relation:-
Let A be any set. A relation R on set A is said to be an antisymmetric relation iff (if and only if)
(a, b) ∈ R and (b, a) ∈ R ⇒a = b for all a, b ∈ A
NOTE It follows from this definition that if (a, b) ∈ R but (b, a) ∉ R then also R is an antisymmetric relation.
Example
-
The identity relation on a set A is an antisymmetric relation
Equivalence relation:-
A relation R on a set A is said to be an equivalence relation on A iff it is
(i) reflexive i.e. (a, a) ∈ R for all a ∈ A.
(ii) symmetric i.e. (a, b) ∈ R (b, a) ∈ R for all a,b ∈ A.
and, (ii) transitive i.e. (a, b) ∈ R and (b, c) ∈ R ⇒(a, c) ∈ R for all a, b, c ∈ A.
An equivalence relation R defined on a set A partitions the set A into pair wise disjoint subsets. These subsets are called equivalence classes determined by relation R.The set of all elements of A related to an element a ∈ A is denoted by [a] i.e. [a] ={ x ∈A : (x, a) ∈ R. This is an equivalence class. Corresponding to every element in A there is an equivalence class. Any two equivalence classes are either identical or disjoint. The collection of all equivalence classes forms a partition of set A.