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Mathematics LibreTexts

7.1.E: Problems on Intervals and Semirings

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    32341
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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).
    \(\text { [Hint: Use Theorem } 2 .]\)

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