 Hello and welcome to the session. Let's discuss the following problem today. Show that the relation r in the set a is equal to 1, 2, 3, 4, 5 given by r is equal to a, b such that mod of a-b is even is an equivalence relation. Show that all the elements of 1, 3, 5 are related to each other and all the elements of set 2, 4 are related to each other. But no element of set 1, 3, 5 is related to any element of set 2, 4. Now let us write the solution. We have a is equal to 1, 2, 3, 4, 5 and the relation r is equal to a, b such that mod of a-b is even. Now let us check for reflexivity. For any a belongs to a, mod of a-a is equal to 0 which is even. Therefore, a, a belongs to r for all a belongs to a. Therefore, r is reflexive. Now let us check for symmetry. Let a, b belongs to r then mod of a-b is even which implies mod of b-a is even which implies b, a belongs to r therefore r is symmetric. Now let us check for transitivity. Let a, b belongs to r and b, c belongs to r then a-b is even and mod of b-c is even which is possible if a and b both are even or both are odd. And b and c both are even or both are odd. So, two cases arise. Case 1, when b is even then a, b belongs to r and b, c belongs to r which implies mod of a-b is even and mod of b-c is even which implies a is even and c is even because b is even which implies mod of a-c is even which implies a, c belongs to r. Now the second case when b is odd. Now take a, b belongs to r and b, c belongs to r which implies mod of a-b is even and mod of b-c is even which implies a is odd and c is odd because b is odd which implies mod of a-c is even which implies a, c belongs to r therefore we have a, b belongs to r and b, c belongs to r which implies a, c belongs to r therefore r is transitive and hence r is an equivalence relation. Now the second part of our question. All the elements of set 1, 3, 5 are related to each other since difference of any two odd numbers is always an even natural number. For example, mod of 1-5 is even similarly elements of set 2, 4 are related to each other since difference of any two even natural numbers is an even natural number. For example, mod of 2-4 or mod of 4-2 is even. Also no element of set 1, 3, 5 is related to any element of set 2, 4 since the difference of an odd and even natural number is not even. For example, mod of 1-4 is equal to 3 which is odd. I hope you understood this problem. Bye and have a nice day.