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# 7.1.E: Problems on Intervals and Semirings

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## Exercise $$\PageIndex{1}$$

Complete the proof of Theorem 1 and Note 1.

## Exercise $$\PageIndex{1'}$$

Prove Theorem 2 in detail.

## Exercise $$\PageIndex{2}$$

Fill in the details in the proof of Corollary 1.

## Exercise $$\PageIndex{2'}$$

Prove Corollary 2.

## Exercise $$\PageIndex{3}$$

Show that, in the definition of a semiring, the condition $$\emptyset \in \mathcal{C}$$ is equivalent to $$\mathcal{C} \neq \emptyset$$.
$$\left.\text { [Hint: Consider } \emptyset=A-A=\cup_{i=1}^{m} A_{i}\left(A, A_{i} \in \mathcal{C}\right) \text { to get } \emptyset=A_{i} \in \mathcal{C} .\right]$$

## Exercise $$\PageIndex{4}$$

Given a set $$S,$$ show that the following are semirings or rings.
(a) $$\mathcal{C}=\{\text { all subsets of } S\}$$;
(b) $$\mathcal{C}=\{\text { all finite subsets of } S\}$$;
(c) $$\mathcal{C}=\{\emptyset\}$$;
(d) $$\mathcal{C}=\{\emptyset \text { and all singletons in } S\}$$.
Disprove it for $$\mathcal{C}=\{\emptyset \text { and all } t w o-p o i n t \text { sets in } S\}, S=\{1,2,3, \ldots\}$$.
In $$(a)-(c),$$ show that $$\mathcal{C}_{s}^{\prime}=\mathcal{C} .$$ Disprove it for $$(\mathrm{d})$$.

## Exercise $$\PageIndex{5}$$

Show that the cubes in $$E^{n}(n>1)$$ do not form a semiring.

## Exercise $$\PageIndex{6}$$

Using Corollary 2 and the definition thereafter, show that volume is additive for $$\mathcal{C}$$ -simple sets. That is,
$\text { if } A=\bigcup_{i=1}^{m} A_{i}(\text {disjoint}) \text { then } v A=\sum_{i=1}^{m} v A_{i} \quad\left(A, A_{i} \in \mathcal{C}_{s}^{\prime}\right) .$

## Exercise $$\PageIndex{7}$$

Prove the lemma for $$\mathcal{C}$$-simple sets.
$$\text { [Hint: Use Problem } 6 \text { and argue as before. }]$$

## Exercise $$\PageIndex{8}$$

Prove that if $$\mathcal{C}$$ is a semiring, then $$\mathcal{C}_{s}^{\prime}(\mathcal{C} \text { -simple sets })=\mathcal{C}_{s},$$ the family of all finite unions of $$\mathcal{C}$$ -sets (disjoint or not).
$$\text { [Hint: Use Theorem } 2 .]$$