 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, consider a binary operation star or n where n is the set of all natural numbers defined as a star b is equal to a cube plus v cube. Choose the correct answer. a is star both associative and commutative. b is star commutative but not associative. c is star associative but not commutative. d is star neither commutative nor 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 a b belonging to a. A binary operation star from a cross a to a is set to be associative if a star b star c is equal to a star bracket b star c for all a b c belonging to a. Let us now start with the solution. We know binary operation star on n is defined as a star b is equal to a cube plus b cube. We will find out if star is commutative. We know a star b is equal to a cube plus b cube for every a b belonging to n where n is the set of all natural numbers. Now, we will find b star a that is equal to b cube plus a cube for every a b belonging to set of all natural numbers. We know from key idea that star is commutative if a star b is equal to b star a. So, we can see a cube plus b cube is equal to b cube plus a cube. This implies a star b is equal to b star a. Now, since a star b is equal to b star a, therefore, star is commutative. So, we can write star is commutative on set n where n is the set of all natural numbers. Now, we will find if star is associative. First of all, let us find out a star b star c it is equal to a cube plus b cube star c which is further equal to a cube plus b cube whole cube plus c cube. So, we can write a star b star c is equal to a cube plus b cube whole cube plus c cube. Now, we will find a star bracket b star c which is equal to a star bracket b cube plus c cube which is further equal to a cube plus b cube plus c cube whole cube. So, we can write a star bracket b star c is equal to a cube plus b cube plus c cube whole cube for every a b c belonging to set of natural numbers. But, we can see a cube plus b cube whole cube plus c cube is not equal to a cube plus b cube plus c cube whole cube. Therefore, we can write a star b star c is not equal to a star bracket b star c. From key idea, we know a star is associative if a star b star c is equal to a star b star c for all a b c belonging to set a. Now, here we can see a star b star c is not equal to a star bracket b star c. So, star is not associative on n. So, we know star is commutative but not associative. So, b is the correct answer. So, the required answer is b. This completes the session. Hope you understood the session. Take care and goodbye.