 Hi and welcome to the session. My name is Shashi and I am going to help you with the following question. The question says determine whether or not each of the definition of star given below is the binary operation. In the event that star is not a binary operation, give justification for this. Fourth part is on z plus where z plus is the set of all positive integers. Define star by a star b is equal to modulus of a minus b. Let us now start the solution. We are given a star b is equal to modulus of a minus b. Now we have to find if star is a binary operation on set of positive integers. So, first of all we will write for all a b belonging to set of positive integers. Modulus of a minus b will always be positive. How this implies? Modulus of a minus b will belong to set of positive integers as modulus of a minus b will always be positive. So, it will always belong to set of positive integers. Therefore, a star b is equal to modulus of a minus b belongs to set of positive integers. So, we can write therefore star is a binary operation on set of positive integers. The answer is yes. This completes the session. Hope you understood the session. Goodbye.