# Show that the relation R defined by R = {(a, b): a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.

We have,

R = {(a,b) : a–b is divisible by 3; a, b Z}

To prove : R is an equivalence relation

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Let a Z

a – a = 0

a – a is divisible by 3

( 0 is divisible by 3).

(a, a) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let a, b Z and (a, b) R

a – b is divisible by 3

a – b = 3p(say) For some p Z

–( a – b) = –3p

b – a = 3 × (–p)

b – a R

R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Let a, b, c Z and such that (a, b) R and (b, c) R

a – b = 3p(say) and b – c = 3q(say) For some p, q Z

a – c = 3 (p + q)

a – c = 3 (p + q)

(a, c) R

R is transitive

Since, R is reflexive, symmetric and transitive

R is an equivalence relation.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 0152 mins
Range of Functions58 mins
Quick Revision of Types of Relations59 mins
Battle of Graphs | various functions & their Graphs48 mins
Functions - 0648 mins
Functions - 1156 mins
Some standard real functions61 mins
Functions - 0947 mins
Quick Recap lecture of important graphs & functions58 mins