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Exercise 32 - Chapter 2

Let \( S \) be a set and let \( \mathbb{P}(S) \) be its powerset, the set of all subsets of \( S \), including the empty subset, \( \emptyset \subseteq S \), and the “everything” subset, \( S \subseteq S \). We can give \( \mathbb{P}(S) \) an order: \( A \le B \) is given by the subset relation \( A \subseteq B \), as discussed in Example 1.40. We propose a symmetric monoidal structure on \( \mathbb{P}(S) \) with monoidal unit \( S \) and monoidal product given by intersection \( A \cap B \).

Does it satisfy the conditions of Definition 2.2?

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Comments

  • 1.

    Let \(R \in P(S)\)

    a) For unitarity, we need to show that \(R \cap S = S \cap R = R\)

    Since \(R \subseteq S\), we know that \(r \in R \Rightarrow r \in S \) so \(\cap\) does nothing

    b) For monotonicity, we need to show that \( A \subseteq C \land B \subseteq D \Rightarrow A \cap C \subseteq B \cap D \)

    From the left side of the implication, we get

    $$a \in A \Rightarrow a \in C$$ $$b \in B \Rightarrow b \in D$$ So \(a, b \in A \cap B \Rightarrow a, b \in C \cap D\) which is equivalent to the right of the implication by the extension axiom of sets

    c) and d) Associativity and commutativity

    $$A \cap (B \cap C) = (A \cap B) \cap C$$ $$A \cap B = B \cap A$$ Follows from the definition of \(\cap\) (consider a Venn diagram of 3 and 2 circles respectively)

    Comment Source:Let \\(R \in P(S)\\) a) For unitarity, we need to show that \\(R \cap S = S \cap R = R\\) Since \\(R \subseteq S\\), we know that \\(r \in R \Rightarrow r \in S \\) so \\(\cap\\) does nothing b) For monotonicity, we need to show that \\( A \subseteq C \land B \subseteq D \Rightarrow A \cap C \subseteq B \cap D \\) From the left side of the implication, we get $$a \in A \Rightarrow a \in C$$ $$b \in B \Rightarrow b \in D$$ So \\(a, b \in A \cap B \Rightarrow a, b \in C \cap D\\) which is equivalent to the right of the implication by the extension axiom of sets c) and d) Associativity and commutativity $$A \cap (B \cap C) = (A \cap B) \cap C$$ $$A \cap B = B \cap A$$ Follows from the definition of \\(\cap\\) (consider a Venn diagram of 3 and 2 circles respectively)
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