MHB Velleman problem 5(d) section 7.2

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The discussion revolves around proving that the cardinality of the power set of positive integers is equivalent to the set of functions from positive integers to the power set of positive integers. The user demonstrates their approach by establishing that the power set of positive integers is similar to the set of functions from positive integers to a binary set. They further apply established cardinality equivalences to show that the function set remains consistent under transformations. The conclusion drawn is that the original statement holds true, and a participant confirms the validity of the proof. The proof effectively illustrates the relationships between these sets and their cardinalities.
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Hi I have to prove

\[ ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\;\sim\; \mathcal{P}(\mathbb{Z^+}) \]

here is my attempt. I have proven that \( \mathcal{P}(\mathbb{Z^+})\;\sim\; ^{\mathbb{Z^+}}\{0,1\} \). Also I am going to use the fact that
if \( A\;\sim B \) and \( C\;\sim D \) then \( ^{A}C\;\sim ^{B}D \). So we get

\[ ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\;\sim\; ^{\mathbb{Z^+}}(^{\mathbb{Z^+}} \{0,1\} ) \]

Also I have proven that for any sets A,B,C we have \( ^{(A\times B)}C\;\sim\; ^{A}( ^{B}C) \). So

\[ ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\;\sim\; ^{(\mathbb{Z^+}\times \mathbb{Z^+} )} \{0,1\} \]

Since \( \mathbb{Z^+}\times \mathbb{Z^+}\;\sim \mathbb{Z^+} \) and \( \{0,1\}\;\sim \{0,1\} \) , we have

\[ ^{(\mathbb{Z^+}\times \mathbb{Z^+} )} \{0,1\}\;\sim\; ^{\mathbb{Z^+}} \{0,1\} \]

So it follows that
\[ ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\;\sim\; ^{\mathbb{Z^+}} \{0,1\} \]

since \( \mathcal{P}(\mathbb{Z^+})\;\sim\; ^{\mathbb{Z^+}}\{0,1\} \) , we get

\[ ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\;\sim\; \mathcal{P}(\mathbb{Z^+}) \]

Is it ok ?

(Emo)
 
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Yes, I think this is fine.
 
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