Question

: Let A = {1, 2, 3, 4, 5, 6, 7}, and let B be the set of all subsets of A. Then B (which
is called the power set of A) is a set whose elements are sets. In particular, note that ∅ and A are both
elements of B (since ∅ ⊆ A and A ⊆ A).
(a) Give an example of two distinct elements B1 and B2 of B that satisfy 0 < n(B1) = n(B2) < 7.
(b) Create a decision algorithm to determine n(B).
Hint. Consider an arbitrary B ∈ B. Imagine how you might “construct” this set B, noting that,
for each x ∈ A, there are 2 alternatives: Either x ∈ B or x /∈ B. Now, think about how many
(distinct) sets can be made using this procedure.
(c) Let C = {x1, ..., xk}, where k is an arbitrary positive integer, and let D be the set of all subsets of
C. Generalize the algorithm you created in part (b) to determine n(D).
(The answer, of course, will be an expression involving k).

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