 Hi and welcome to the session. My name is Shashi and I am going to help you to solve the following question. Question says, for each primary operation star, define below. You can find whether star is commutative or associative. Fifth part is on z plus, where z plus is the set of all positive integers. Define a star b is equal to a raised to the power b. Let us now start the solution. We are given on z plus, where z plus is the set of positive integers. Minority operation star is defined as a star b is equal to a raised to the power b. Now we know for commutative binary operation, a star b is equal to b star a. Now we know a star b is equal to a raised to the power b. It is given in the question and b star a would be equal to b raised to the power a. a raised to the power b is not equal to b raised to the power a. So therefore, a star b is not equal to b star a. So star is not commutative on z plus, plus is the set of all positive integers. Now we know for associative binary operation, a star bracket b star c is equal to a star b star c. Now we know a star bracket b star c is equal to a star b raised to the power c, which is further equal to a raised to the power b raised to the power c. Now let us find out the right hand side that is a star b star c. Now this is equal to a raised to the power b star c, which is further equal to a raised to the power b, whole raised to the power c. Now applying the law of exponents we get a raised to the power bc. This is because we know the law of exponents x raised to the power m, whole raised to the power n is equal to x raised to the power mn. Now clearly we can see a raised to the power b raised to the power c is not equal to a raised to the power bc. So therefore we get a star bracket b star c is not equal to a star b star c. This implies binary operation star is not associative all z plus. So our required answer is binary operation star is neither commutative nor associative. This completes the session. Hope you understood the session. Take care and good bye. Thank you.