Let Z be the set of all integers and R be the relation on Z defined as R={(a,b):a,b∈Z,and (a-b)is divisible by 5}. Prove that R is an equivalence relation.

SOLUTION: It is given that 
		R={(a,b):a,b∈Z,and (a-b)is divisible by 5}
	R is reflexive, as a-a=0=0×5    ∀ a∈Z
		⇒	a-a is divisible by 5
		⇒	(a,a)∈R
	R is symmetric, as (a,b)∈R     where a,b∈Z
		⇒	a-b is divisible by 5
		⇒	a-b=5p  for some integer p
		⇒	b-a=5(-p)
		⇒	b-a is divisible by 5
		⇒	(b,a)∈R
	R is transitive, as (a,b)∈R     where a,b∈Z
			a-b is divisible by 5
		⇒	a-b=5p  for some integer p
	For (b,c)∈R  where b,c∈Z
			b-c is divisible by 5
		⇒	b-c=5q  for some integer q
			(b,c)∈R     where b,c∈Z
		Now 	(a-b)+(b-c)=5p+5q
		⇒	a-c=5(p+q)
		⇒	a-c is divisible by 5.       						 
                ⇒	(a,c)∈R
		Thus, 	(a,b)∈R, (b,c)∈R							 
               ⇒	(a,c)∈R   ∀ a,b,c∈Z
	Since R is reflexive, symmetric and transitive
	Therefore, R is equivalence on Z.   	  Hence Proved