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# Exercise 32 - Chapter 2

edited June 2018

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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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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