 Hi, and welcome to the session. I am Shashi. Let us do our question. Question is, let star B binary operation on set Q of rational numbers as follows. Second part is, A star B is equal to A square plus B square. Find 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. You know, a binary operation star on 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 set 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 A square plus B square. Now, first of all, we will check if the given operation is commutative. Now, we know A star B is equal to A square plus B square for every AB belonging to Q. Let us find out B star A. We know B star A is equal to B square plus A square for every AB belonging to Q, right? We know A square plus B square is equal to B square plus A square as addition is commutative for every AB belonging to Q implies A star B is equal to B star A. So, we can write binary operation star is commutative on Q where Q is the set of rational numbers. Now, let us check if star is associative. First of all, we will find A star bracket B star C we know it is equal to A star B square plus C square. We know A star B is equal to A square plus B square. So, B star C is equal to B square plus C square. Now, this is further equal to A square plus B square plus C square whole square, right? So, we get A star bracket B star C is equal to A square plus B square plus C square whole square. Now, let us find out A star B star C it is equal to A square plus B square star C this is further equal to A square plus B square whole square plus C square. So, we get A star B star C is equal to A square plus B square whole square plus C square. We know A square plus B square plus C square whole square is not equal to A square plus B square whole square plus C square. So this implies A star bracket B star C is not equal to A star B star C. Therefore, we can write binary operation star is not associated on set Q where Q is the set of all rational numbers. So our final answer is binary operation star completes the session. Hope you understood the session. Take care and goodbye.