Challenge Problem. Recall that given a set S, the power set of S, denoted P(S)), is theset of all subsets of S. Similarly, recall that the empty set, denoted (Ø, is the set containingno elements.Let S = {rabbit}. Then, define P'(S) = P(S), the power set of S; and for integersn > 2, let P"(S) = P(P"-'(S)) be the n-th power set of S. Give, via the roster method,the setA = {n € N| {0} e P"(S)}.

Let S=rabbit. Then, P1S=PS and PS=∅,S. Therefore, P1S=∅,S. Given P2S=PP1S. Now, PP1S=∅,S,∅,P1S…

Answered: Challenge Problem. Recall that given a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Challenge Problem. Recall that given a set S, the power set of S, denoted P(S)), is the
set of all subsets of S. Similarly, recall that the empty set, denoted (Ø, is the set containing
no elements.
Let S = {rabbit}. Then, define P'(S) = P(S), the power set of S; and for integers
n > 2, let P"(S) = P(P"-'(S)) be the n-th power set of S. Give, via the roster method,
the set
A = {n € N| {0} e P"(S)}.

Expert Solution

Step 1

Let $S = \{rabbit\}$ .

Then, $P^{1} (S) = P (S)$ and $P (S) = \{\emptyset, S\}$ .

Therefore, $P^{1} (S) = \{\emptyset, S\}$ .

Given $P^{2} (S) = P (P^{1} (S))$ .

Now, $P (P^{1} (S)) = \{\{\emptyset\}, \{S\}, \emptyset, P^{1} (S)\}$

Hence, $P^{2} (S) = \{\{\emptyset\}, \{S\}, \emptyset, P^{1} (S)\}$

Further,

$\begin{array}{rcl} P^{3} (S) & = & P (P^{2} (S)) \\ = & P (\{\emptyset\}, \{S\}, \emptyset, P^{1} (S)) \\ = & \{\{\emptyset\}\}, \{\emptyset\}, \{\{S\}\}, \{P^{1} (S)\}, \dots \dots \dots \dots \dots \dots \dots \end{array}$

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