 Hi and welcome to the session. I am Shashi. Let us do one question. Question is, let star be a binary operation on set Q of rational numbers as follows. First part is A star B is equal to A minus E. 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. 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 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 now start with the solution on set Q where Q is the set of all rational numbers. Binary operation star is defined as is equal to A minus B. First of all, let us check if star is commutative. We know A star B is equal to A minus B and B star A would be equal to B minus A but A minus B is not equal to B minus A for every AB belonging to set of rational numbers that is Q. So, this implies A star B is not equal to B star A. Therefore, we can write binary operation star is not commutative on set Q where Q is the set of all rational numbers. Now, let us check if star is associative. First of all, let us find A star B star C is equal to A minus B star C which is further equal to A minus B minus C. So, A star B star C is equal to A minus B minus C. Now, let us find out A star bracket B star C is equal to A star B minus C which is further equal to A minus B minus C or we can say A minus B plus C. So, we get A star B star C is equal to A minus B plus C. Here we can see A minus B minus C is not equal to A minus B plus C. Now, this implies A star B star C is not equal to A star bracket B star C. Therefore, we can write binary operation star is not associative on Q where Q is the set of all rational numbers. So, our final answer is star is neither commutative nor associative. Please, in the session, hope you understood the session. Take care and goodbye.