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Exercise 47 - Chapter 2

Start with a preorder \( (P, \le) \), and use it to define a \(\textbf{Bool}\)-category as we did in Example 2.44. In the proof of Theorem 2.46 we showed how to turn that Bool-category back into a preorder. Show that doing so, you get the preorder you started with.

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  • 1.
    edited May 23

    Bool \( := (\mathbb{B}, \le, \tt{false}, \wedge) \).

    The following is not the complete proof, it only shows the mapping of data and not that the rules of the two concepts are honored.

    Given a preorder \( \mathcal{P} : (P, \le) \) use it to define, map to, a Bool-category \( \mathcal{Q} \).

    The objects of \( \mathcal{P} \) are the objects of \( \mathcal{Q} \).

    For every pair of objects \( (x, y) \) we assign an element of \( \mathbb{B} = \{false, true\} \): simply assigning \(\tt{true}\) if \(x \le y \), and \(\tt{false} \) otherwise.

    Given a Bool-category \( \mathcal{Q} \) use it to define, map to, a preorder \( \mathcal{P} \).

    The objects of \( \mathcal{Q} \) are the objects of \( \mathcal{P} \).

    For every pair of objects \( (x, y) \) we observe the value of its element, \( \mathbb{B} = \{false, true\} \): simply add the arrow in \( \mathcal{P} \) when \(\tt{true}\) skipping the \(\tt{false}\) pairs.

    Comment Source:**Bool** \\( := (\mathbb{B}, \le, \tt{false}, \wedge) \\). The following is not the complete proof, it only shows the mapping of data and not that the rules of the two concepts are honored. Given a preorder \\( \mathcal{P} : (P, \le) \\) use it to define, map to, a **Bool**-category \\( \mathcal{Q} \\). The objects of \\( \mathcal{P} \\) are the objects of \\( \mathcal{Q} \\). For every pair of objects \\( (x, y) \\) we assign an element of \\( \mathbb{B} = \\\{false, true\\\} \\): simply assigning \\(\tt{true}\\) if \\(x \le y \\), and \\(\tt{false} \\) otherwise. Given a **Bool**-category \\( \mathcal{Q} \\) use it to define, map to, a preorder \\( \mathcal{P} \\). The objects of \\( \mathcal{Q} \\) are the objects of \\( \mathcal{P} \\). For every pair of objects \\( (x, y) \\) we observe the value of its element, \\( \mathbb{B} = \\\{false, true\\\} \\): simply add the arrow in \\( \mathcal{P} \\) when \\(\tt{true}\\) skipping the \\(\tt{false}\\) pairs.
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