Textbook Problem

In Example 7.2.8 a one-to-one correspondence was de?ned from the power set of {a, b} to the set of all strings of 0’s and 1’s that have length 2. Thus the elements of these two sets can be matched up exactly, and so the two sets have the same number of elements.

a. Let $X=\left\{x_1,x_2,...,x_n\right\}$ be a set with n elements. Use Example 7.2.8 as a model to define a one-te-one correspondence from P(X), the set of all subsets of X, to the set of a1l strings of 0’s and 1’s that have length n.

b. In Section 9.2 we show that there are $2^n$ strings of 0’s and 1’s that have length n. What does this allow you to conclude about the number of subsets of P(X)? (This provides an alternative proof of Theorem 6.3.1.)