 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. Fourth part is A star B is equal to A minus B whole square. Find which of the binary operations are commutative and which are associative. First of all, let us know 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 V star A for every A V belonging to X. Now, let us define when binary operation star is associative. A binary operation star from A cross A to A is said to be associative if A star V star C is equal to A star bracket V star C for every A V belonging to set A. Let us start with the solution now. We know on set Q that is the set of all rational numbers, binary operation star is defined as A star B is equal to A minus B whole square. Now, first of all, we will find out if star is commutative. We know A star V is equal to A minus B whole square. Now, we can write it as A square plus B square minus 2 AB. We know A minus B whole square is equal to A square plus B square minus 2 AB. So, we get A star B is equal to A square plus B square minus 2 AB. Now, let us find out B star A. We know B star A would be equal to B minus A whole square which would be further equal to B square plus A square minus 2 AB. So, we can write B star A is equal to B square plus A square minus 2 BA. Now, we know A square plus B square minus 2 AB is equal to B square plus A square minus 2 BA. We know multiplication is commutative and addition is also commutative. So, LHS and RHS both are equal. So, this is equal to A minus B whole square and this is equal to B minus A whole square. So, we get A minus B whole square is equal to B minus A whole square. This further implies A minus B whole square is equal to A star B. So, we substitute A star B here and B minus A whole square is equal to B star A. So, we get A star B is equal to B star A. So, therefore, we can write binary operation star is commutative on Q where Q is the set of all rational numbers. Now, let us check if star is associative. First of all, let us find out A star B star C. We know this is equal to A minus B whole square star C. This is further equal to A minus B whole square minus C whole square. We know A star B is equal to A minus B whole square. So, this term is our A. So, A minus C is our B minus C whole square, right? Now, let us find out A star bracket B star C. We know it is equal to A star B minus C whole square. We know B star C would be equal to B minus C whole square. Now, this is further equal to A minus B minus C whole square whole square, right? But A minus B whole square minus C whole square is not equal to A minus B minus C whole square whole square. So, this implies A star B star C is not equal to A star B star C, right? So, we can write binary operation star is not associative on set Q, where Q is the set of all rational numbers. So, our final answer is binary operation star is commutative on set Q, where Q is the set of all rational numbers. This completes the session. Hope you understood the session. Take care and goodbye.