7.1.E: Problems on Intervals and Semirings

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).
$\text { [Hint: Use Theorem } 2 .]$