 Hello and welcome to the session. Let us understand the following question today. Determine whether the following relation is reflexive, symmetric and transitive. Given to us relation r in the set A equal to 1, 2, 3 so on till 13, 14 defined as r is equal to x, y such that 3x minus y is equal to 0. Before writing the solution let us define the relation. A relation r in a set A is called reflexive if a comma a belongs to r for every a belongs to a. Symmetric if a1 comma a2 belongs to r which implies a2 comma a1 belongs to r. For all a1 comma a2 belongs to a and transitive if a1 comma a2 belongs to r and a2 comma a3 belongs to r which implies a1 comma a3 belongs to r. For all a1 comma a2 comma a3 belongs to a. Now this is the key idea behind our question. Now let us proceed with the solution. Given to us as a relation r is equal to x comma y such that 3x minus y is equal to 0 where x comma y belongs to a is equal to the elements 1 comma 2 so on till 13 comma 14. Now first of all let us find the elements of r. Given to us as 3x minus y is equal to 0 so now for x is equal to 1 it implies y is equal to 3. Similarly for x is equal to 2 it implies y is equal to 6 for x is equal to 3 it implies y is equal to 9 for x is equal to 4 we have y is equal to 12 for x is equal to 5 we have y is equal to 15 therefore we have but here 15 does not belongs to a. As we can see that a is defined with elements till 14 therefore we have r as 1 comma 3, 2 comma 6, 3 comma 9 and 4 comma 12. Now let us check for reflexivity. We know 1 belongs to a but 1 comma 1 does not belongs to r. Here we can see that 1 comma 1 is not there in r so therefore r is not reflexive. Now let us check for symmetry. Here we can see that 1 comma 3 belongs to r but 3 comma 1 does not belongs to r. Now since we have studied in the key idea that if a1 comma a2 belongs to r which implies a2 comma a2 should also belongs to r this is the symmetric property but here it does not hold therefore r is not symmetric. Now let us check for transitivity. We have 1 comma 3 belongs to r 3 comma 9 belongs to r but 1 comma 9 does not belongs to r therefore r is not transitive therefore we can see that r is not reflexive r is not symmetric and r is not transitive therefore the required answer is neither reflexive nor symmetric nor transitive. I hope you understood the question and enjoyed the session. Bye and have a nice day.