Introduction
If you have a basic knowledge about the relations and sets and their different properties, you are good to go. Here in this blog post, I will be discussing the Equivalence of Relations with pertinent details. At the end of this blog, everything will be clear to you.
So, without further ado, let’s start the topic.
Definition
A relation on a set is an equivalence relation if the relation is reflexive, symmetric, and transitive. Got confused about what is reflexive, symmetric and transitive properties?
Let’s first discuss each one of them.
Reflexive
A relation R on a set A is reflexive if (a, a) ∈ R for every element a ∈ A.
Too many mathematical terms, right?
Okay, let me simplify it,
Consider a set A = {1 , 2 , 3 }.
Let’s define all possible elements in a relation(Cartesian product) in the form of a matrix.
A X A = 
Now, the relations that contain the diagonal elements of the above matrix are considered the Reflexive relations.
R = 
So, R = {(1,1),(2,2),(3,3)}.
Example
Let the set be A = {1, 2, 3}
Let’s define some of the relations on the set,
R1 = {(1,1),(2,2),(3,3)}
R2 = {(1,1),(2,2),(1,3),(2,3)}
R3 = {(1,1),(2,2),(3,3),(1,3),(2,3)}
Now, First, try to identify the Reflexive relations by yourself.
If your answer is R1 and R3, then you are right.
You might be thinking, why R3, right?
See, We are only concerned about the (a, a) ∈ R for all a ∈ A. If any additional (a,b) ∈ R where a,b ∈ A then also the Relation would be Reflexive.
I think now you got it.
Let’s now move to the Symmetric Property.
Symmetric
A relation R on a set A is Symmetric if (b, a) ∈ R whenever (a, b) ∈ R for all a, b ∈ A.
Here also we are considering the same set A = {1,2,3}
So, according to the definition, if (1,2) ∈ R, then (2,1) must also belong to R.
Let’s understand it in terms of a matrix,
Let’s R = {(1,2),(1,1),(2,1),(2,3),(3,2)}
The matrix representation of the relation is,
R = 
RT = 
If R = RT, then only the relation is symmetric.
Example
Let’s define some of the relations over the set A = {1, 2, 3}
R1 = {(1,1),(1,2),(2,1),(2,2)}
R2 = {(1,1),(1,2),(2,2),(2,3)}
R3 = {(2,3),(3,2),(1,2)}
Try to identify the symmetric relations.
If your answer is only R1, then you are correct.
For, R1
(1,1) is present
(1,2) is present, and (2,1) is also present.
(2,2) is present
For, R2
(1,2) is present but not (2,1)
(2,3) is present but not (3,2)
For, R3
(1,2) is present but not (2,1)
I hope it is clear to you.
Let’s now move to the Transitive property,
Transitive
A Relation R on a set is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a,c) ∈ R for a,b,c ∈ A.
Now for the set A = {1 , 2, 3}
If (1, 2) ∈ R and (2,3) ∈ R then (1,3) ∈ R.
Example
Let’s define some of the relations over the set A = {1, 2, 3}
R1 = {(1,1),(1,2),(2,1),(2,2)} Transitive
R2 = {(1,1),(1,2),(2,2),(2,3)} Transitive
R3 = {(2,3),(3,2),(1,2)} Not Transitive (2, 3) ∈ R and (3, 2) ∈ R but (2, 2) ∉ R.
So, a binary relation on a set is an Equivalence Relation If it is Reflexive, Symmetric and Transitive.
So, let a,b,c ∈ A.
If (a,a) ∈ R (Reflexive).
If (a,b) ∈ R if and only if (b ,a) ∈ R ( Symmetric).
If (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R (Transitive)
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