 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, define below, determine whether star is commutative or associative. Third part is, on Q, where Q is the set of all rational numbers, define A star V is equal to AB upon 2. First of all, let us understand the key idea to solve the given question. We know a binary operation star on set A is commutative if A star V 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. On Q, we have binary operation star defined as A star B is equal to AB upon 2. Now, first of all, we will find if star is commutative. We know from Q, idea star is commutative if A star B is equal to B star A. So, we know A star B is equal to AB upon 2. Now, we will find B star A which is equal to BA upon 2, right? Now, we know AB upon 2 is equal to BA upon 2. As multiplication is commutative, so AB upon 2 must be equal to BA upon 2. This implies A star B is equal to B star A. This further implies that star is commutative on set Q where Q is the set of all rational numbers. Now, we will check if star is associative. From key idea, we know star is associative if A star B star C is equal to A star bracket B star C. First of all, we will find A star B star C which is equal to AB upon 2 star C. We know A star B is equal to AB upon 2, so we will substitute AB upon 2 in place of A star B and get AB upon 2 star C. Now, this is further equal to ABC upon 2 multiplied by 2, right? We know A star B is equal to A multiplied by B upon 2, so we get ABC upon 2 multiplied by 2 which is further equal to ABC upon 4. So, A star B star C is equal to ABC upon 4, right? Now, we will find A star bracket B star C is equal to A star BC upon 2. B star C would be equal to BC upon 2. Now, this would be further equal to ABC upon 2 multiplied by 2 which is further equal to ABC upon 4. So, we get A star bracket B star C is equal to ABC upon 4. Now, we can see A star B star C is equal to ABC upon 4 and A star bracket B star C is also equal to ABC upon 4. So, this implies A star B star C is equal to A star bracket B star C for every ABC belonging to Q. Let us name this equation as 1 and this equation as 2. Now, we can write from equation 1 and equation 2, we get A star B star C is equal to A star B star C for every ABC belonging to Q. Therefore, we get star is associative on set Q where Q is the set of all rational numbers. So, our final answer is star is both commutative and associative. This completes the session. Hope you understood the session. Have a nice day.