 Hello and welcome to the session. In this session we discuss the following question which says proof that the relation r in the set a equal to 1, 2, 3, 4, 5 given by r equal to a b such that modulus of a minus b is even is an equivalence relation. Before we move on to the solution, let's see when a relation is said to be equivalence relation. First of all let's consider let a be a non-empty set then a relation r on the set a is said to be first reflexive if we have a a belongs to r for each a belongs to the set a. Then the relation r is said to be symmetric if a b belongs to r implies that b a belongs to r for all a and b belongs to a. Then the relation r is said to be transitive if we have a b belongs to r b c belongs to r implies that a c belongs to r for all a b c belongs to a. Then the relation r in set a is an equivalence relation if it is reflexive symmetric and transitive. This is the key idea that we use for this question. Let's proceed with the solution now. We are given a set a equal to the set with an events 1 2 3 4 5 and we have a relation r defined in set a as a b such that modulus of a minus b is even. We have to show that r is an equivalence relation for that we have to show that it is reflexive symmetric and transitive. So first let's check the reflexivity of the relation r. We take let a be the element of the set a that is a belongs to set a then modulus of a minus a is equal to 0 which is even therefore we say that a a belongs to the set a and therefore the relation r is reflexive. Next we check if the relation r is symmetric or not. For that we have to show that if a b belongs to r then b a also belongs to r where a and b are the elements of the set a. So for this let a and b be the elements of set a and a b belongs to r. So this means modulus of a minus b is even which means modulus of minus of a minus b is even that is modulus of b minus a is even and thus we say that b a belongs to r. Now for a b belongs to r we get b a belongs to r and therefore relation r is symmetric. Let's see if the relation r is transitive or not. For this we take let a b and c be three elements of the set a and let a b belongs to r b c belongs to r. So this means modulus of a minus b is even and modulus of b minus c is even. This means a minus b is equal to plus minus 2k1 and b minus c is equal to plus minus 2k2 where we have k1 and k2 belongs to n that is the set of natural numbers. So further a minus b plus b minus c is equal to plus minus 2k1 plus minus 2k2 or you can say this is equal to plus minus 2 into k1 plus minus k2 or you can say a minus c is equal to plus minus 2 into k1 plus minus k2 where this k1 and k2 belongs to n that means we have modulus of a minus c is even and therefore ac belongs to r thus relation r is transitive. As we have shown that the relation r is reflexive, symmetric and transitive is an equivalence relation. So we have shown that r is an equivalence relation. So this completes the equation. Hope you have understood the solution of this question.