 Hi and welcome to the session I am Shashi and I am going to help you to solve the following question. Question says for each binary operation star defined below determine whether star is commutative or associated. Fourth part is on z plus that is the set of positive integers define A star B is equal to 2 raised to the power A multiplied by B. Let us now start the solution on z plus star is defined as A star B is equal to 2 raised to the power P B. First of all let us check if the given binary operation is commutative for that we will find B star A. B star A is equal to 2 raised to the power B A. We know for commutative binary operation A star B is equal to B star A. Now we know A star B is equal to 2 raised to the power A B and B star A is equal to 2 raised to the power B A and we also know that multiplication is commutative. So, we can write 2 raised to the power A B is equal to 2 raised to the power B A for all A B belonging to z plus. So, we can write therefore A star B is equal to B star A this implies star is commutative on z plus where z plus is the set of all positive integers. Let us now check if the given binary operation is associated. We know for associative binary operation A star bracket B star C is equal to A star B star C. Now, let us find out A star bracket B star C. Now, we know B star C is equal to 2 raised to the power B C. So, we get A star 2 raised to the power B C. Now, this is further equal to 2 raised to the power A multiplied by 2 raised to the power B C. Let us find out A star B star C this is equal to 2 raised to the power A B star C. Now, this is further equal to 2 raised to the power 2 raised to the power A B multiplied by C. Really, we can see these two terms are unequal. So, we can write A star bracket B star C is not equal to A star B star C thus star is not associative on z plus. So, our required answer is binary operation star is commutative but not associative. This completes the session. Hope you understood the session. Take care and goodbye.