 Hello and welcome to the session. In this session we discuss the following question which says consider the relation R defined on the set A equal to x belongs to N such that x is greater than equal to 1 and less than equal to 9 given by R is equal to ordered pair AB such that modulus A minus B is even shown that R is an equivalence relation. What is the set of elements related to 3? Before we move on to the solution let's recall that when a relation in a set is said to be an equivalence relation. So let's consider a relation R in a set A. So let's see when a relation R in a set A is reflexive, symmetric and transitive. A relation R in a set A is reflexive belongs to capital R for every A belongs to capital A and the relation R in a set A is symmetric. A1 A2 belongs to R implies that A2 A1 belongs to R for all A1 A2 belongs to capital A that is A1 A2 are the elements of the set A and the relation R in a set A is transitive A2 belongs to R A2 A3 belongs to R implies that A1 A3 belongs to R for all A1 A2 A3 belongs to the set A. So we have got the conditions when a relation R in a set A is reflexive, symmetric and transitive. Now a relation R is said to be an equivalence relation the relation R is reflexive, symmetric and transitive. So this is the key idea that we use for this question. Now let's see the solution. So in the question we are given a set A which is defined as X belongs to N such that X is greater than equal to 1 and less than equal to 9 then we have a relation R which is defined as ordered pair AB such that modulus of A minus B is even. Now we have to show that R is an equivalence relation so for that we have to show that R is reflexive, symmetric and transitive. Suppose let's see reflexivity for this let's consider an element small a which belongs to the set A. Consider the key idea in this we have that a relation R in a set A is reflexive if the ordered pair AA belongs to R where A is the element of the set A. So we have taken A to be the element of set A now consider modulus of A minus A now this would be equal to 0 which is even so this means that the ordered pair AA belongs to R where we have this A is the element of set A and therefore we say that the relation R is reflexive. Now let's check whether the relation R is symmetric or not we know that the relation R is symmetric if A1 A2 belongs to R implies that A2 A1 also belongs to R where A1 and A2 are the elements of set A. We consider let the ordered pair AB belongs to the relation R where we have A and B are the elements of the set A. Now since the ordered pair AB belongs to R so this would mean the modulus of A minus B is even this is by the definition of the set R or the relation R. Now further we have modulus of minus of A minus B is also even which means that modulus of B minus A is even and this gives us that ordered pair BA belongs to R where we have AB are the elements of set A. So when the ordered pair AB belongs to R this implies the ordered pair BA belongs to R where A and B are the elements of set A therefore the relation R is symmetric. Now let's check the transitivity of the relation R from the key idea we have that the relation R in a set A is transitive if A1 A2 belongs to R and A2 A3 belongs to R this implies that A1 A3 belongs to R and A1 A2 A3 are the elements of set A. So let's take ordered pair AB belongs to R and the ordered pair BC also belongs to R where AB and C are the elements of set A. So as ordered pair AB belongs to R so this would mean modulus of A minus B is even and as ordered pair BC belongs to R so modulus of B minus C is even or you can say that A minus B would be equal to plus minus 2N1 and B minus C is equal to plus minus 2N2 for some N1 N2 belongs to N. Further we have A minus B plus B minus C is equal to plus minus 2 N2 N1 plus minus N2 or you can say we have A minus C is equal to plus minus 2 N2 N for some N belongs to N where we have taken this N that is the small N as N1 plus minus N2 so this means modulus of A minus C is equal to 2N for some N belongs to N that is modulus of A minus C is even as it is the multiple of 2. So from the definition of the relation R we get the ordered pair AC belongs to R. So we have if ordered pair AB belongs to R and the ordered pair BC belongs to R so this implies the ordered pair AC belongs to R and therefore we say that the relation R is transitive. Thus we find that the relation R is reflexive symmetric transitive hence we say that the relation R is an equivalence relation. In the question we have that what is the set of elements related to 3. Now the set of elements to 3 is given as B belongs to the set A such that modulus of 3 minus B is even and so this set would be given by the set of elements 1 3 5 7 9. So this is the set of elements related to 3. So we have shown that the relation R is an equivalence relation and also we have found out the set of elements related to 3. So this completes the session hope you have understood the solution of this question.