 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 binary operation star, define below. Determine whether star is commutative or associative. First part is on z, where z is the set of integers. Define a star b is equal to a minus b. First of all let us know that a binary operation on set x is commutative. a star b is equal to b star a for every a b belonging to set x. Also a binary operation star on set a is said 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 set a. So clearly we can see a binary operation on set x is commutative if a star b is equal to b star a and binary operation is associative on set a if a star b star c is equal to a star bracket b star c. This is the key idea to solve the given question. Let us now start the solution. We are given on z that is the set of integers. Binary operation star is defined as a star b is equal to a minus b. First of all let us find out if the binary operation star is commutative. For that we will first of all find out b star a that is equal to b minus a. Now we know for commutative binary operation b star a is equal to a star b a minus b is not equal to b minus a. For all a b belonging to set that is the set of integers. So therefore star is what a commutative binary operation on set we can write a star b is not equal to b star a. Now let us check if the binary operation star is associative 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 b star c is equal to a star b minus c as b star c is equal to b minus c. Now this is further equal to a minus b minus c. So we get a star bracket b star c is equal to a minus bracket b minus c. Similarly we get a star b star c is equal to a minus b minus c. We know a minus b minus c is not equal to a minus b minus c for all a b c belonging to z. The binary operation star is not as associative on z. So our final answer is star is neither commutative nor associative. This completes the session. Hope you understood the session. Goodbye.