 Hello and welcome to the session. In this session we will discuss the following question and the question says let P is equal to the set containing the elements 3, 6, 9 and Q is equal to the set containing the elements 3, 7. Find A, P cross P, B, P cross Q, C, Q cross P, D, Q cross Q. Before we start solving the question let us first recall what is Cartesian product. Cartesian product of two non-empty sets is the set of all ordered pairs whose first components are selected from elements of the first set, second components are selected from elements of second set. So this is our key idea for this question and using this key idea we shall solve the question. Let's start the solution now. We are given set P is equal to the set containing the elements 3, 6, 9 and set Q is equal to the set containing the elements 3, 7. In the first part we have to find what is P cross P. Now using our key idea P cross P is the set of all ordered pairs whose first components are selected from elements of the first set which is P in this case and second components are selected from elements of second set which is also P. So P cross P is equal to the set containing the ordered pairs. First we will fix the first component as 3 and pair it with all the elements of the second set which is P again. So the first ordered pair is 3, 3. Now in the second ordered pair the first component is 3 again and second component is 6. Similarly in the third ordered pair the first component is 3 again and the second component is 9. So in this way we have first fixed the first component as 3 and paired it with all the elements of the second set which is P. Now we will fix the first component as 6 and pair it with all the elements of the set P. So we get the ordered pair 6, 3, ordered pair 6, 6 and ordered pair 6, 9. Now we fix the first component as 9 and pair it with all the elements of the set P. So we get the ordered pair 9, 3, the ordered pair 9, 6 and the ordered pair 9, 9. So in this way we have found the set P cross P which is our answer for the first part. Now in the second part we have to find what is P cross Q. P cross Q is the set containing the ordered pairs whose first components are elements of the set P and second components are elements of the set Q. So P cross Q is equal to the set containing the ordered pair whose first component is 3 which is an element of the set P and second component is 3 which is an element of the set Q. We will first fix the first component as 3 and pair it with all the elements of the set Q. So the second ordered pair is 3, 7. Next we will fix the first component as 6 and pair it with the elements of set Q. So we get the ordered pair 6, 3 and the ordered pair 6, 7. Now fixing the first component as 9 and pairing it with the elements of set Q we get the ordered pairs 9, 3 and the ordered pair 9, 7. So in this way we have found P cross Q which is our answer for the second part. In the third part we have to find Q cross P. Now Q cross P is the set containing ordered pairs whose first components are elements of the set Q and second components are elements of the set P. So we will fix each element of the set Q and pair it with all the elements of the set P. Thus Q cross P is equal to the set containing the ordered pair 3, 3, ordered pair 3, 6, ordered pair 3, 9. So we have fixed the first component as 3 which is an element of the set Q and paired it with the elements 3, 6, 9 which are elements of the set P. Similarly fixing the first component as 7 we get the ordered pair 7, 3, ordered pair 7, 6 and ordered pair 7, 9. Thus we have found Q cross P which is the answer for the third part. In the fourth part we have to find Q cross Q. Now Q cross Q is the set containing the ordered pairs whose first components are elements of the set Q and second components are elements of the set Q. So Q cross Q is equal to the set containing the ordered pair 3, 3, ordered pair 3, 7, ordered pair 7, 3 and ordered pair 7, 7. As done above we have first fixed each element of the set Q individually and paired it with the elements of the second set which is Q in this case. So we get Q cross Q as this set which is our answer for the fourth part. With this we end our session. Hope you enjoyed the session.