A relation defined on a set A is called equivalence relation, if it is reflexive, symmetric and transitive. In mathematics, we have many relations that are reflexive, symmetric and transitive. This is the most important relation in math.
A relation R is said to be an equivalence relation if it is reflexive, symmetric and transitive. So, we can say if,

1. (a, b) $\in$ R, $\forall$ a, b ∈ A.
2. (a, b) $\in$ R $\Rightarrow$ (b, a) $\in$ R, for all a, b ∈ A.
3. (a, b) $\in$ R and (b, c) $\in$ R $\Rightarrow$ (a, c) $\in$ R, $\forall$ a, b, c $\in$ A.

## Equivalence Relation Proof

An equivalence relation divides a set into disjoint subsets. These disjoint subsets are called equivalence classes. All elements of these classes are equal to themselves, not with any element from the different classes.

If we have a set A and an equivalence relation on A, then for any element a $\in$ A, the equivalence classes is the subset of all element in A which are equivalent to a. To prove equivalence relation, we have to prove that the given relation is reflexive, symmetric and transitive. Then, automatically that relation becomes an equivalence relation.

### Solved Example

Question: If A be the set of all lines in a plane and R be the relation in A defined
as R = {(A1, A2) : A1 is parallel to A2}. Prove that R is an equivalence relation.
Solution:

Reflexive: Let we have line A1 and we know that a line is parallel to itself, hence A1 II A1 $\in$ R.
Symmetric: If A1II A2, then A2 II A1 i.e. if A1 is parallel to A2, then A2 is
parallel to A1. So, the relation R is symmetric.
Transitive: Let A1, A2 and A3 be three lines in such a way that
A1 II A2 $\in$ R, A2 II A3 $\in$ R
$\Rightarrow$ A1 II A3$\in$ R
Hence, relation R is the equivalence relation.

## Equivalence Relation Examples

Let us learn about equivalence relation with the help of the following examples.

### Solved Examples

Question 1: If A = {1,2,3,4,5,6,7,8} be the given set and a relation R is defined on A as R = { (x, y) : both are even}, then show that R is an equivalence relation.
Solution:
Given R = { (x,y) : both are even}.
Reflexive: Since x$\in$ A, then x is even so (x,x) $\in$R. Hence, R is reflexive.
Symmetric: Let (x,y) $\in$ R $\Rightarrow$ both x and y are even. So, we say (y,x) $\in$ R. Therefore, R is symmetric.
Transitive: Let x, y, z $\in$ A such that (x, y)$\in$ R, (y, z)$\in$ R
$\Rightarrow$ (x, z) $\in$ R.
If all the three conditions are satisfied, then R is an equivalence relation.

Question 2: Show that "=" is an equivalence relation.
Solution:
To prove that "=" is an equivalence relation, we have to prove that "=" is reflexive, symmetry and transitive.
1. a = a is true for all a. Hence, "=" is reflexive.
2. If a = b, then b = a (obvious), so "=" is symmetry.
3. If a = b and b = c, we can say a = c for all a, b and c. So, "=" is transitive
Hence "=" is an equivalence relation.