 Hi and welcome to the session. Let us discuss the following question. Question says, determine whether or not each of the definition of star, even below, gives a binary operation. In the event that the star is not a binary operation, give justification for this. First part is on z plus, where z plus is the set of all positive integers, define star by a star b is equal to a minus b. Let us now start the solution. We are given a star b is equal to a minus b. We have to determine if star is a binary operation on all set of positive integers. Now, here two cases are possible. Let us take case one, where a is greater than b. And a is greater than b, then a star b is equal to a minus b. Now, a star b is equal to a minus b, would be a positive integer. Or we can say a minus b would be greater than 0. Now, this implies a minus b belongs to set of positive integers. Let us take the case two. Case two is when a is less than b. Now, when a is less than a, then we can write a star b equal to a minus b would be less than 0. So, here a minus b does not belongs to set of positive integers. For example, if we find 3 star 7, it is equal to 3 minus 7, which is further equal to minus 4, which is less than 0. Or we can say minus 4 does not belong to set of positive integers. So, we can write therefore, star is not a binary operation on z plus. Where z plus is the set of all positive integers. So, our final answer is no. This completes the session. Hope you understood the session. Take care and good bye.