Suppose we have the preorders
\[ \mathcal{X} := \left[ \text{monoid} \rightarrow \text{category} \leftarrow \text{preorder} \right] \]
\[ \mathcal{Y} := \left[ \text{nothing} \rightarrow \text{this book} \right] \]

1. Draw the Hasse diagram for the preorder \\( \mathcal{X}^{op} \times \mathcal{Y} \\).
2. Write down a profunctor \\( \Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \\) and, reading \\( \Lambda(x, y) = \text{true} \\) as
“my uncle can explain \\(x\\) given \\(y\\)”, give an interpretation of the fact that the preimage of
\\( \text{true} \\) forms an upper set in \\( \mathcal{X}^{op} \times \mathcal{Y} \\).

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**Definition 4.2**

Let \\( \mathcal{X} = (X, \le_X ) \text{ and } \mathcal{Y} = (Y, \le_Y ) \\) be preorders.
A _feasibility relation_ for \\( \mathcal{X} \\) given \\( \mathcal{Y} \\) is a monotone map
\[ \tag{4.3} \Phi : \mathcal{X}^{op} \times \mathcal{Y} \rightarrow \textbf{Bool} \]

We denote this by \\( \Phi : \mathcal{X} \nrightarrow \mathcal{Y} \\).
Given \\( x \in X \text{ and } y \in Y \\), if \\( \Phi(x, y) = \text{true} \\) we say \\(x\\) _can be obtained given_ \\(y\\).