EQUIVALENCE RELATION

About "Equivalence relation"

Equivalence relation :

As we have rules for reflexive, symmetric and transitive relations, we don't have any specific rule for equivalence relation.

Let R be a relation defined on a set A. 

If the three relations reflexive, symmetric and transitive hold in R, then it is an equivalence-relation.  

To verify equivalence, we have to check whether the three relations reflexive, symmetric and transitive hold.

Apart from the equivalence relation, we have some other relations on sets.

Let us discuss the other different types of relations here:

The other different types of relations on sets are 

(i) Reflexive relation

(ii) Symmetric relation

(iii) Transitive relation

(iv) Identity relation

(v) Inverse relation

Let us discuss the above different types of relations in detail.

Reflexive relation

The rule for reflexive relation is given below. 

" Every element is related to itself "

Let R be a reflexive relation defined on the set A and

 a ∈ A, then R  =  { (a, a) / for all a ∈ A }

That is, every element of A has to be related to itself. 

Example :

Let A  =  {1, 2, 3 } and R be a reflexive relation defined on  set A.

Then, R  =  { (1, 1), (2, 2), (3, 3) } 

Symmetric relation

Let R be a symmetric relation defined on the set A and 

a, b ∈ A, then R  =  { (a, b), (b, a) / for all a, b ∈ A }

That is, if "a" is related to "b", then "b" has to be related to "a" for all "a" and "b" belonging to A. 

Example :

Let A be the set of two male children in a family, R be a relation defined on set A and R  =  "is brother of". Verify whether R is symmetric. 

Let a, b ∈ A.  

If "a" is brother of "b", then "b" has to be brother of "a".

Clearly, R  =  { (a, b), (b, a) }

Hence, R is symmetric. 

Transitive relation

Let R be a transitive relation defined on the set A and          a, b, c ∈ A, 

then R  =  { (a, b), (b, c), (a, c) / for all a,b,c ∈ A }

That is,

If "a" is related to "b" and "b" is related to "c", then "a" has to be related to "c".  

Example :

Let A  =  { 1, 2, 3 }, R be a relation defined on  set A as "is less than" and R  = { (1, 2), (2, 3), (1,3) } Verify R is transitive. 

From the given set A, let a = 1, b = 2 and c = 3.

Then, we have

(a, b )  =  (1, 2) --> 1 is less than 2 

(b, c )  =  (2, 3) --> 2 is less than 3 

(a, c )  =  (1, 3) --> 1 is less than 3 

That is, if 1 is less than 2 and 2 is less than 3, then 1 is less than 3

Clearly, the above points prove that R is transitive. 

Important note :

For a particular ordered pair in R, if we have (a, b) and we don't have (b, c), then we don't have to check transitive for that ordered pair. 

So, we have to check transitive, only if we find both (a, b) and (b, c) in R. 

If the above three relations reflexive, symmetric and transitive hold, then it is equivalence relation.

Identity relation

The rule for identity relation is given below.

" Every element is related to itself only "

Let R be an identity relation defined on the set A and

 a ∈ A, then R  =  { (a, a) / for all a ∈ A }

That is, every element of A has to be related to itself only. 

Example :

Let A  =  {1, 2, 3 } and R be an identity relation defined on  set A.

Then, R  =  { (1, 1), (2, 2), (3, 3) }

Here, every element of A is related to it self and not to any other different element. 

Inverse relation

Let R be a relation defined on the set A, a, b ∈ A and           R  =  { (a, b) }, then the inverse relation of R is

R⁻¹ = { (b, a) }

In the given relation, if "a" is related to "b", then in the inverse relation "b" will be related to "a". 

Example :

Let R  =  { (1, 2), (2, 3), (7, 5) }

Then R⁻¹ =  { (2, 1), (3, 2), (5, 7) }

Difference between reflexive and identity relation

Reflexive = " Every element is related to itself "

Identity   = " Every element is related to itself only "

Let us consider an example to have better understanding of the difference between the two relations. (Reflexive vs Identity)

Let A = {1, 2, 3}

Let R₁ and R₂ be two relations defined on set A such that

R₁ = { (1,1) , (2,2) , (3,3) , (1,2) }

R₂ = { (1,1) , (2,2) , (3,3) }

When we look at R₁ , every element of A is related to itself and also, the element "1" is related to a different element "2".

More details about R₁

(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"

(ii) Apart from "1" is related to "1", "1" is also related to "2"

Here we can not say that "1" is related to "1" only. Because "1" is related to "2" also.   

This is the point which makes the reflexive relation to be different from identity relation.

Hence R₁ is reflexive relation

When we look at R₂, every element of A is related to it self and no element of A is related to any different element other than the same element.

More details about R₂

(i) "1" is related to "1", "2" is related to "2" and "3" is related to "3"

(ii) "1" is related to "1" and it is not related to any different element. 

     The same thing happened to "2" and "3".

(iii) From the second point, it is very clear that every element of R is related to itself only. No element is related to any different element 

This is the point which makes identity relation to be different from reflexive relation.

Hence R₂ is identity relation

Important terms related to relations 

If a relation does mapping from the set A to set B, then we can define the following terms. 

Domain : Set A

Co domain : Set B

Range : Elements of B involved in mapping. 

Let R and R⁻¹ be the two relations which are inverse to each other.  

Then we have,

Domain (R)  =  Range (R⁻¹)

Range (R)  =  Domain (R⁻¹)

After having gone through the stuff given above, we hope that the students would have understood "Equivalence relation". 

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