A short Note on Equivalence Relation

We all have studied the identicalness of fractions in our earlier classes. Here, many fractions that didn’t look similar had a parallelism association with each other. Likewise is the concept and idea for the equivalence relation which provides the detail that functions don’t look similar, but are related to each other in some way or the other. An Equivalence Relation refers to a category of binary relation which has all the properties of relation including symmetry, transitivity, and reflexiveness. Now, you would surely be reckoning about the above-mentioned relations. These relations provide distinct aspects to the relation that exists between the two functions or more than that too. 

What is Relation in mathematics? 

In mathematics, Relation refers to an established association between two sets of evidence or variables. Sets to be considered as associated should have components that are associated with each other. 

With relation to the properties of relation and functions, if two sets are related to each other, set A has a relation with set B, then the conclusion which can be drawn is that the values of set A are regarded as the domain and the values of set B as the range. 

What is an Equivalence Relation?

An Equivalence Relation refers to a category of binary relation which has all the properties of relation including symmetry, transitivity, and reflexiveness. 

1)Symmetric property

Let us take two elements from a set, say t and u, and say that (t, u) belongs to the relation. If we want our relation to be symmetric, then (u, t) should also belong to the relation. 

              (t, u) ∈ R, then (u, t) ∈ R

For example: If we have a set X and it has elements {4,5,6,}, and the Relation has (4,5) as ordered pairs, then the set would be only called symmetric provided the relation should also have (5,4). Nevertheless, if the relation has (4,6) as pairs but doesn’t have (6,4) in it, then in that case the relation wouldn’t be called a symmetric relation. Unlike the reflexive relation, here we don’t consider all the elements, but any element of the set. If (5,6) is not there in the relation, then it is not necessary that (6,5) should be there in the relation, without that too, the relation can be called a symmetric relation. 

2) Reflexive property

We call any relation reflexive, provided you take an element from the given set X then the ordered pair (g, g) belongs to that relation. This should be for all the elements in that set. 

               g ∈ X, then  (g, g) ∈ R

For example: If we have a set X and it has elements {4,5,6,}, then the set would be only called reflexive when 

                          {4,4} ∈ R  

                          {5,5} ∈ R

                          {6,6} ∈ R

If even one of them doesn’t lie in the relation, then the set cannot be called reflexive. So, if an element says g is related to itself, then it would be called a reflexive relation. 

3) Transitive property

Talking of relation being transitive, this also goes for any element in the relation, and not every element of the set should be included in the relation. Suppose we take (c, d, e, f) from the given set and (c, d) and (d, e) lie in the relation. Then the relation would be called transitive, if (c, e) is also present in the relation. 

           If (c, d) ∈ R and (d, e) ∈ R 

                 then, (c, e) ∈ R

For example: If we have a set X and it has elements {4,5,6,} and relation as (4,5) and (5,6) which belongs to the set R then the set would be only called transitive when (4,6) is also a part of the relation. Unlike the reflexive relation, here we don’t consider all the elements, but any element of the set and if something is excluded it is not necessary to have an ordered pair of that too in the relation.            

So, if a relation has all three of these properties, then it would be referred to as an Equivalence relation. 

EXAMPLE QUESTION ON EQUIVALENCE RELATION

QUESTION: Suppose there is a set Q = {5,6,7,8 } that is bestowed by the relation R that states {(m, n): |m-n| is even}, then prove that the above relation mentioned is equivalence.

ANSWER: If a relation has all three properties of reflexiveness, transitive and symmetric, then it would be referred to as an Equivalence relation. 

Reflexive 

For the case of reflexive, 

                   m ∈ R, then (m, m) ∈ R

Here, in this example |m-m| = 0 is an even number. Therefore, the relation convinces the property of reflexiveness. 

Symmetry

For the case of symmetry, 

                  (m, n) ∈ X, then  (n, m) ∈ R

Here, in this example, |m-n| is equal to |n-m| and the result is even. Therefore, the condition of symmetry is also satisfied in this case.

Transitive

For the case of transitivity, 

                If (m, n) ∈ R and (n, o) ∈ R 

                 then, (m, o) ∈ R

Here, in this example, as |m-n| and |n-o| is even, and then |m-o| is also an even number, then the case is transitive too.

As all the relations are satisfied, this can be said as an Equivalence relation. 

Conclusion

Thus, an Equivalence Relation refers to a category of binary relation which has all the properties of relation including symmetry, transitivity, and reflexiveness. So, if all the properties are satisfied, we can conclude the relation as an equivalence relation. One such concept is also called the Equivalence Class that refers to a subset G of H in such a manner that the components m and n which pertain to the relation cannot exist outside the subset G in any case.