 Hi and welcome to the session, I am Shashi and I am going to help you to solve the following question. Question is, for each binary operation star defined below, determine whether star is commutative or associative. Second part is, on Q, where Q is the set of all rational numbers, define A star B is equal to AB plus 1. First of all, let us understand the key idea to solve the given question. We know a binary operation star on the set A is commutative if A star B is equal to B star A for every AB belonging to set A. Now a binary operation star from A cross A to A is set to be associative if A star B star C is equal to A star bracket B star C for every ABC belonging to set A. Let us now start with the solution. Below, a binary operation star defined as A star B is equal to AB plus 1. First check if star is commutative. We know A star B is equal to AB plus 1. So we can write B star A is equal to BA plus 1. Now we know A multiplied by B is equal to B multiplied by A as multiplication is commutative. AB plus 1 is equal to BA plus 1, A star B is equal to B star A. From key idea, we know star is commutative if A star B is equal to B star A for every AB belonging to Q. So we can write star is commutative on set Q. The Q is the set of all rational numbers. From key idea, we know star is associative when A star B star C is equal to A star bracket B star C. First of all, let us find out A star B star C which is equal to AB plus 1 star C which is further equal to AB plus 1 multiplied by C plus 1, right? So we find A star B star C is equal to ABC plus C plus 1. Let us now find out A star bracket B star C which is equal to A star BC plus 1, right? Which is further equal to A multiplied by BC plus 1 plus 1 which is further equal to ABC plus A plus 1. So we get A star bracket B star C is equal to ABC plus A plus 1. Now we can see ABC plus C plus 1 is not equal to ABC plus A plus 1. So we can write, this implies A star B star C is not equal to A star bracket B star C for every ABC belonging to Q. Hence, star is not associative on Q. This is the set of all rational numbers. So our required answer is star is commutative but not associative. This completes the session. Hope you understood the session. Take care and goodbye.