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Here are four puzzles from Lecture 9:

**Puzzle 22.** What operation on subsets corresponds to the logical operation "not"? Describe this operation in the language of posets, so it has a chance of generalizing to other posets. Based on your description, find some posets that *do* have a "not" operation and some that don't.

**Puzzle 23.** Suppose \(f : X \to Y \) is a function between sets. For any \( S \subseteq X \) define

$$ f_{!}(S) = \{y \in Y: \; y = f(x) \textrm{ for some } x \in S\} $$ and

$$ f_{\ast}(S) = \{y \in Y: x \in S \textrm{ for all } x \textrm{ such that } y = f(x)\} . $$ I showed in the lecture that \( f_{\ast}: P(X) \to P(Y) \) is the left adjoint of the inverse image map \( f^{\ast}: P(Y) \to P(X) \). Show that \( f_{\ast}: P(X) \to P(Y) \) is the right adjoint of this map.

**Puzzle 24.** Let \( X \) be the set of states of your room, and \( Y \) the set of states of a thermometer in your room: that is, thermometer readings. Let \(f : X \to Y \) map any state of your room to the thermometer reading. What is \(f_{!}(\{\text{there is a living cat in your room}\})\)? How is this an example of the "liberal" or "generous" nature of left adjoints, meaning that they're a "best approximation from above"?

**Puzzle 25.** What is \(f_{\ast}(\{\text{there is a living cat in your room}\})\)? How is this an example of the "conservative" or "cautious" nature of right adjoints, meaning that they're a "best approximation from below"?

I believe these were all answered in the lengthy discussion of Lecture 9, so if you get stuck browse through that!

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