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: $f:X\in\mathcal{P}(S)\rightarrow min(P\backslash\pi(X))\in P.$
- Determine $f(\mathcal{P}(S))$;
- For each $a \in f(\mathcal{P}(S))$ determine a $X\in\mathcal{P}(S)$ such that $f(X) = a$
- characterize the elements $X\in\mathcal{P}(S)$ such that $f(X)=5$
- 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:
- $f(\mathcal{P}(S))=\{2,3,5\}$
- $2\rightarrow\{1\}; \qquad 3\rightarrow\{2\}; \qquad 5\rightarrow\{1,2,3\}$
- $\{\{1,2,3,4\}, \{1,2,3\}, \{2,3\}, \{3,4\}, \{2,3,4\}, \{1,3,4\}\}$
- $[\emptyset]_{\sim}=\{a:f(a)=f(\emptyset)=2\}=\{\emptyset, \{1,3\}, \{1\},\{3\}\}=[\{1,3\}]_{\sim}=[\{1\}]_{\sim}=[\{3\}]_{\sim}$
Is this right?