Use mathematical induction to show that a set with n elements that 2n subsets. This is a dicult problem in that it is a simple problem to solve. Simplicity can sometimes be dicult to spot. Herere some helpful notes for this problem. Lets consider the set A = {a1,a2}. It has two elements and therefore we expect 4 subsets. Indeed, the subsets of A are: , {a1}, {a2}, and {a1,a2} Now consider A0 = {a1,a2,a3}. This set has 3 elements, therefore we expect it to have 8 subsets. Its subsets are: , {a1}, {a2}, {a1,a2}, {a3}, {a1,a3}, {a2,a3}, {a1,a2,a3} There is a particularly interesting property between the rst half of the subsets of A0 and the second half. If you can identify this property and exploit it, you can solve this problem. It may also be helpful to remember that X = X.

(e) (3 bonus points) Use mathematical induction to show that a set with n elements that 2n subsets. This is a difficult problem in that it is a simple problem to solve. Simplicity can sometimes be difficult to spot. Here're some helpful notes for this problem. Let's consider the set A -[ai, a2l. It has two elements and therefore we expect 4 subsets. Indeed, the subsets of A are: , a, a2), and fai,a2] Now consider A, {al, a2, ). This set has 3 elements, therefore we expect it to have 8 subsets. Its subsets are: e, {al), {az), {al, az), (as), {al, as), {a2, as), {al, a2, a3} There is a particularly interesting property between the first half of the subsets of A' and the second haf If you can identify this property and exploit it, you can solve this problem. It may also be helpful to remember that UX-X