In discovering the expression for finding the number of subsets of a set with n element...

In discovering the expression for finding the number of subsets of a set with n elements, we observed that for the first few values of n, increasing the number of elements by one doubles the number of subsets. Here, you can prove the formula in general by showing that the same is true for any value of n. Assume set A has n elements and s subsets. Now add one additional element, say e, to the set A. (We now have a new set, say B, with n + 1 elements.) Divide the subsets of B into those that do not contain e and those that do.

(a) How many subsets of B do not contain e? (Hint: Each of these is a subset of the original set A.)

(b) How many subsets of B do contain e? (Hint: Each of these would be a subset of the original set A, with the additional element e included.)

(c) What is the total number of subsets of B?

(d) What do you conclude?