 Hello and welcome to the session. In this session we will discuss the following question and the question says, how many subsets do the following sets have? List them as directed. First part is, A is equal to letters of the word Chicago. List all proper subsets with one element. Second part is, the set A is equal to the set containing the elements 5, 10, 15. List all proper subsets with two elements. Third part is, the set B is equal to two digit numbers between 5 and 13. List all proper subsets with three elements. Fourth part is, the set P is equal to the set containing the elements A, B, C, D. List all proper subsets with five elements. Before we start solving the question, let us first recall how many subsets a given set has. If a set has n elements, then the number of subsets of the set is two raised to the power n. Now let us review what is a proper subset, the subset of a given set except the set itself is called a proper subset. So this is the key idea for this question and using this key idea we will solve the question. Let's start the solution now. The first part is, A is equal to letters of the word Chicago. We will write the set A in roster form since the repetition of the elements is not allowed. So set A is equal to the set containing the elements C, H, I, A, G, O. The number of subsets of A is equal to two raised to the power 6. Since there are six elements in the set A which is equal to 64, now we have to find all proper subsets with one element. So it's proper subsets with one element set containing the element C, set containing the element H, set containing the element I, set containing the element A, set containing the element G and set containing the element O. So this is our answer for the first part. Now the second part is, set A is equal to the set containing the elements 5, 10, 15. The number of subsets of A is equal to two raised to the power 3. Since there are three elements in the set A, this is equal to 8, we have to find all proper subsets with two elements. Proper subsets containing two elements are the set containing the elements 5, 10, set containing the elements 10, 15 and set containing the elements 5, 15. This is our answer for the second part. The third part is, the set B is equal to two digit numbers between 5 and 13. Numbers between 5 and 13 are 6, 7, 8, 9, 10, 11 and 12. Now from these the two digit numbers are 10, 11 and 12. So the set B in roster form can be written as the set containing the elements 10, 11, 12. The number of subsets of B is equal to two raised to the power 3. Since there are three elements in the set B, this is equal to 8. Now we have to find all proper subsets with three elements. The only subset of the set B having three elements is the set B itself. Since the set itself cannot be a proper subset, so there exists no proper subset of B which has three elements. This is our answer for the third part. Now the fourth part is set P is equal to the set containing the elements A, B, C, D. The number of subsets of P is equal to two raised to the power 4. Since there are four elements in the set P, this is equal to 16. We have to find all proper subsets with five elements. Since the set P itself contains four elements, so there cannot exist any proper subset of the set P having five elements. So we can say that as the set itself has four elements, therefore proper subset having five elements is none. This is our answer for the fourth part. With this we end our session. Hope you enjoyed the session.