# Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given byR = {(a, b) : |a – b| is even} is an equivalence relation.

In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.

Given that, a, b A, R = {(a, b) : |a – b| is even}.

Now,

R is Reflexive if (a,a) R a A

For any a A, we have

|a-a| = 0, which is even.

(a,a) R

Thus, R is reflexive.

R is Symmetric if (a,b) R (b,a) R a,b A

(a,b) R

|a-b| is even.

|b-a| is even.

(b,a) R

Thus, R is symmetric .

R is Transitive if (a,b) R and (b,c) R (a,c) R a,b,c A

Let (a,b) R and (b,c) R a, b,c A

|a – b| is even and |b – c| is even

(a and b both are even or both odd) and (b and c both are even or both odd)

Now two cases arise:

Case 1 : when b is even

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

|a – b| is even and |b – c| is even

a is even and c is even [ b is even]

|a – c| is even [ difference of any two even natural numbers is even]

(a, c) R

Case 2 : when b is odd

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

|a – b| is even and |b – c| is even

a is odd and c is odd [ b is odd]

|a – c| is even [ difference of any two odd

natural numbers is even]

(a, c) R

Thus, R is transitive on Z.

Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functional Equations - JEE with ease48 mins
Drift velocity & its relation with currentFREE Class
Relation between electric field and potential34 mins
Relation between Electric field & Potential45 mins
Interactive Quiz on Drift velocity & its relation with current38 mins
Interactive quiz on Relation between electric field and potential43 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses