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