 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, let star be a binary operation on set Q of rational numbers as follows. Fifth part is A star B is equal to AB upon 4. Fine, which of the binary operations are commutative and which are associative? First of all, let us understand the key idea to solve the given question. We know a binary operation star on set X is commutative if A star B is equal to B star A for every AB belonging to set X. Now, a binary operation star from A cross A to A is said 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 start with the solution now. We know on set Q binary operation star is defined as A star B is equal to AB upon 4. First of all, let us find out if star is commutative. Now, for every AB belonging to set Q, we have A star B is equal to AB upon 4, right? Now, we will find out A star A. We know B star A would be equal to BA upon 4. Now, we know that AB upon 4 is equal to BA upon 4 since multiplication is commutative, right? So, this implies A star B is equal to B star A. Now, since A star B is equal to B star A, this implies binary operation star is commutative on Q. So, we can write binary operation star is commutative on set Q where Q is the set of all rational numbers. Now, let us check if binary operation star is associated. Now, we know for every ABC belonging to set Q, where Q is the set of all rational numbers, A star B star C is equal to AB upon 4 star C. Now, this is further equal to AB upon 4 multiplied by C upon 4. We know this bracket is our A and this C is our B. We know A star B is equal to A multiplied by B upon 4 which is further equal to ABC upon 4 multiplied by 4 or we can write ABC upon 16. So, we get A star B star C is equal to ABC upon 16. Now, let us find out A star bracket B star C. We know it would be equal to A star BC upon 4, right? We know B star C is equal to BC upon 4. Now, this would be further equal to A multiplied by BC upon 4 upon 4, right? Now, this is further equal to ABC upon 4 multiplied by 4 which is further equal to ABC upon 16. So, we get A star bracket B star C is equal to ABC upon 16. Now, let us name the previous equation as equation one and this equation as equation two. Now, from equation one and two A star B star C is equal to A star bracket B star C. So, we can write binary operation star is associative on set Q where Q is the set of all rational numbers. So, our final answer is binary operation star is commutative and binary operation star is associative on set Q where Q is the set of all rational numbers. Hope you understood the session. Take care and goodbye.