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Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) \end{aligned}$$ and $$\begin{aligned} f(x)=\{y \in P: y \mid x\} \end{aligned}$$

Let $X=\{1,4,5,8,10\}$ and $f(X)=\{\{\emptyset\}, \{2\}, \{5\}, \{2\}, \{2,5\}\}$. Let $\Sigma$ be a partial order defined as follows:

$$\begin{aligned} x\text{ }\Sigma \text{ } y \Leftrightarrow f(x) \subset f (y) \text{ or } x=y\end{aligned}$$

draw the Hasse diagram relative to $(X, \Sigma)$.

The $f$ function clearly isn't injective, because $f(4) = f(8)=\{2\}$. I am unsure how the Hasse diagram should be drawn: in this case I have a repetition, so do I have to omit one of the elements with the same image element? So is my Hasse diagram correct?

enter image description here

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    You're specifying the domain and range of $f$, but not which elements map to what.2012-07-29
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    I suspect that $\subset$ is the notation for [strict inclusion](http://en.wikipedia.org/wiki/Subset#The_symbols_.E2.8A.82_and_.E2.8A.83). Maybe you could check it in the text from which you took the problem. (BTW I think it is always good to mention the source of the problem in the post - give the name of the book, or a link, if they are available.)2012-09-28

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