 Hi and welcome to the session. Today we will discuss the following question which says first is the binary operation star defined on set N that is the set of natural numbers given by A star B is equal to A plus V upon 2 for all A, B belonging to N commutative. Second is the above binary operation star associated. Before moving on to the solution let's recall that a binary operation star from A cross A to A is said to be commutative if A star B is equal to V star A for all A, B belonging to set A. And 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 V star C for all A, B, C belonging to set A. This is the key idea for this question. Now let's see its solution. Here the binary operation star defined on set N is given by A star B is equal to A plus V upon 2 for all A, B belonging to set N. So star is a binary operation from N cross N to N given Y. Now we need to find out whether the binary operation star is commutative or not. So for that first of all let us find A star B. So A star V will be equal to A plus B upon 2 which can also be written as B plus A upon 2 which will be equal to B star A by the definition of the binary operation star. So we have A star B is equal to B star A for all A, B belonging to set N. Therefore the binary operation star is commutative on the set N. Now let's find out whether the binary operation star is associative or not. So for this first of all let us find out A star B star C. So A star B star C will be equal to A plus B upon 2 star C which will be equal to A plus B upon 2 plus C upon 2. Now this will be equal to A plus B plus 2 C upon 4. Now let us find A star V star C which will be equal to A star B plus C upon 2. Now this will be equal to A plus V plus C upon 2 whole upon 2 that is 2A plus B plus C upon 4. Now A star B star C is equal to A plus V plus 2 C upon 4 and A star B star C is equal to 2A plus V plus C upon 4 and it is very clear that these two are not equal. So we have A plus B plus 2C upon 4 is not equal to 2A plus B plus C upon 4 in general. Thus this implies that A star B star C is not equal to A star V star C. Therefore we can say that the binary operation star is not associative on the set N and with this we finish this session. Hope you must have understood the question. Goodbye take care and have a nice day.