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Let $S=\{1,2,3,4\}$ and $P$ the set of positive prime numbers. If $X \in S$ then $\pi(X) = \{p\in P : \exists x \in X \wedge p|x\}$ the application $f$ is so defined: $$f:X\in\mathcal{P}(S)\rightarrow min(P\backslash\pi(X))\in P.$$

  1. Determine $f(\mathcal{P}(S))$;
  2. For each $a \in f(\mathcal{P}(S))$ determine a $X\in\mathcal{P}(S)$ such that $f(X) = a$
  3. characterize the elements $X\in\mathcal{P}(S)$ such that $f(X)=5$
  4. considering the equivalence relation defined in $\mathcal{P}(S)$ as $X\sim Y \Leftrightarrow f(X) = f(Y)$ determine $[\emptyset]_{\sim}$

The power set is so defined: $$\mathcal{P}(S) = \{\emptyset, \{1,2,3,4\},\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,2,3\},\{2,3\},\{3,4\},\{2,3,4\},\{1,3\},\{1,4\},\{2,4\},\{1,2,4\},\{1,3,4\}\}$$

And these are the images for each family of sets over S: $$\pi(\emptyset) = 2 = \pi(\{1\})=\pi(\{3\})=\pi(\{1,3\})$$ $$\pi(\{2\}) = 3 = \pi(\{4\})=\pi(\{1,2\})=\pi(\{1,4\})=\pi(\{2,4\})=\pi(\{1,2,4\})$$ $$\pi(\{1,2,3,4\}) = 5 = \pi(\{1,2,3\})=\pi(\{2,3\})=\pi(\{3,4\})=\pi(\{2,3,4\})=\pi(\{1,3,4\})$$

and so as far as I know these should be correct answers:

  1. $f(\mathcal{P}(S))=\{2,3,5\}$
  2. $2\rightarrow\{1\}; \qquad 3\rightarrow\{2\}; \qquad 5\rightarrow\{1,2,3\}$
  3. $\{\{1,2,3,4\}, \{1,2,3\}, \{2,3\}, \{3,4\}, \{2,3,4\}, \{1,3,4\}\}$
  4. $[\emptyset]_{\sim}=\{a:f(a)=f(\emptyset)=2\}=\{\emptyset, \{1,3\}, \{1\},\{3\}\}=[\{1,3\}]_{\sim}=[\{1\}]_{\sim}=[\{3\}]_{\sim}$

Is this right?

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@Jack, not everything with sets is set theory. At best this is elementary set theory. –  Asaf Karagila Mar 25 '12 at 19:04
@AsafKaragila: Hmm, I should have noticed that there is a elementary-set-theory tag. Thanks. –  Jack Mar 25 '12 at 19:45
Can someone help me with the real problem? ^_^ –  Mariano Mar 25 '12 at 21:56
The notation in your answer in $2$ is confusing; the arrows should go the other way, I think (I suspect you were thinking "for $2\in f(\mathcal{P}(S))$, we can use $\{1\}$", etc; but the arrows make it seem like you are saying that $2$ maps, under $f$ to $\{1\}$...) In 3, you did not "characterize" the elements that map to $5$, you instead wrote down $f^{-1}(5)$. That's not the same; you want to give a description that tells you what those elements are (e.g., "$X\in f^{-1}(5)$ if and only if blah.") –  Arturo Magidin Mar 25 '12 at 23:49
So how I should've wrote the 3? –  Mariano Mar 26 '12 at 6:31
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