Let set A = {1, 2, 3, 4, 5}. 

And relation on set is defined as R = {(a, b) ∶ |a − b| is even}. 

Reflexivity : Let a ∊ A.

Then, |a − a| = 0 which is an even number. (Because, 0 is divisible by 2, hence 0 is an even number) 

Therefore, (a, a) ∊ R, ∀ a ∊ A. 

Hence, relation is a reflexive relation. 

Symmetricity : Let a,b ∊ A such that (a,b) ∊ R. 

⇒ |a − b| is an even number. 

⇒ |b − a| is an even number. ( \(\because\)|a − b| = |−(b − a)| = |b − a| ) 

⇒ (b, a) ∊R ∀ a,b ∊ A . 

Hence, if (a, b) ∊ R. Then, (b, a) ∊R ∀ a, b ∊ A.. 

Therefore, relation is a symmetric relation. 

Transitivity : Let a,b ,c ∊ A such that (a, b) ∊ Rand (b, c) ∊ R. 

⇒ |a − b| is an even number and |b − c| is an even number. 

⇒ a, b and are odd numbers or a, b and c are even numbers. 

(Because |a − b| and |b − c| are even numbers is only possible when a,b and c are of same type) 

⇒ a and c both are odd numbers or and both are even numbers. 

⇒ |a − c| is an even number. 

⇒ (a, c) ∊ R ∀ a,b ,c ∊ A. 

Hence, if (a, b) ∊ R and (b, c) ∊ R. Then, (a, c) ∊ R ∀ a,b ,c ∊ A.

Therefore, relation is a transitive relation. 

Since, relation is reflexive, symmetric and transitive relation. 

Therefore, relation is an equivalence relation. 

Hence, relation defined on the set = {1,2 ,3 ,4 ,5 }, given by R= {(a, b) ∶ |a − b| } is an equivalence relation.