Let Z be the set of all integers. A relation R is defined on Z by x R y to mean x – y is divisible by 5. Show that R is an equivalence relation on Z.

x – y is divisible by 5 ⟺ x – y = 5 m

where m is an integer

⇒ y – x = 5 (– m)

∴  y R x                                              

∴     R is symmetric on Z.

x – x = 0 = 5. 0  x R x ∀ x ∈ z      

∴ R is reflexive on Z.

Let x R y and y R z

⇒ x – y = 5 m   and  y – z = 5 n              

⇒ x – y + y – z = 5 m + 5 n           

⇒ x – z = 5(m + n)

∴  x R z

∴  R is transitive on Z.  Thus R is an equivalence relation on Z.