 Hi and welcome to the session, I am Asha and I am going to help you with the following question that says, let A, B and C be the sets such that A union B is equal to A union C and A intersection B is equal to A intersection C show that B is equal to C. So first let us learn that if two sets A and B are equal then the number of elements in set A is equal to the number of elements in set B. Also the number of elements in A union B is equal to the number of elements in set A plus number of elements in set B minus number of elements which are in A intersection B. So with the help of these two ideas we are going to solve the above problem. So this is a key idea again with the solution, where we are given A union B is equal to A union C this implies that number of elements in A union B will be equal to the number of elements in A union C. This is our key idea. Again number of elements in A union B can be written as number of elements in A plus number of elements in B minus number of elements in A intersection B is equal to now number of elements in A union C can be written as number of elements in A plus number of elements in C minus number of elements in A intersection C. Now we are also given that A intersection B is equal to A intersection C, so by our key idea this implies that number of elements in A intersection B will be equal to number of elements in A intersection C. And now by using this statement it can further be written as number of elements in A plus number of elements in B minus replacing number of elements in A intersection B where number of elements in A intersection C. On the right hand side we have number of elements in A plus number of elements in C minus number of elements in A intersection C. This further implies that number of elements in B is equal to number of elements in C since these two cancels out. Now let it belongs to A and Y belongs to B. This implies that X, Y belongs to A union B. A union B is equal to A union C. This implies that X, Y belongs to A union C as we can say that X belongs to A and Y belongs to C. From this step it implies that X belongs to A and Y belongs to C. This implies for all X belonging to A and Y belonging to B, Y belongs to C also. And now from this step from here we find that all elements which belong to B belong to C also and also the number of elements in the set B and C are equal. Therefore combining both of these equations we find that set P is equal to C. Hence this completes the solution. Hope you enjoyed it. Take care and have a good day.