 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says let r be a relation from n to n defined by r, the set of all ordered pair a, b such that a, b belongs to n and a is equal to b square. Are the following true? First is the ordered pair a, a belongs to r for all a belonging to n. Second is a, b belongs to r implies b also belongs to r. Third is ordered pair a, b belongs to r, b, c belongs to r implies a, c belongs to r. Justify your answer in each case. Let us now start with the solution and the relation r is defined by the set of all ordered pair a, b such that a, b belongs to n and a is equal to b square. Let us now start with the first part which is ordered pair a, a belongs to r for all a belong to n. Now the relationship between a and b in the relation r is a, b square. We have the ordered pair a, a so on replacing b by a in this ordered pair we get a is equal to a square which is not true except a is equal to 1. And hence pair a, a do not belong to r for all a belonging to n. The answer is no. The ordered pair a do not belong to the relation r. Let us now proceed on to the second part which says the ordered pair a, b belongs to r implies ordered pair b, a belongs to r. Now here we have a, b belongs to r implies a is equal to b square. And this implies that b, a belongs to r implies b is equal to a square. And since a is equal to b square does not imply that b is equal to a square except for a is equal to b is equal to 1. Hence a, b belongs to r do not implies that b, a belong to r. Hence the answer to the second part is no. Let us now proceed on to the third part. Here we have the ordered pair a, b belongs to r, b, c belong to r implies the ordered pair a, c belong to r. Now here we have a, b belongs to r implies a is equal to b square. Also b, c belong to r implies b is equal to c square. And combining these two we find that a is equal to b square and b is c square. So we have c square whole square which is equal to c raised to the power 4. So this implies that a is equal to c raised to the power 4. Now we are given that if these two ordered pair belong to r then we have ordered pair a, c belong to r. This implies a is equal to c square. On observing these two we find that a is equal to c square and a is equal to c raised to the power 4 which is true only if a is equal to c is equal to 1. a, b belongs to r or b, c belongs to r do not imply that a, c belong to r. Hence the answer to the third part is also no. So this completes the solution. Hope you enjoyed it. Take care. Have a good day.