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?