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Mathematics LibreTexts

7.1.E: Problems on Intervals and Semirings

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Exercise 7.1.E.1

Complete the proof of Theorem 1 and Note 1.

Exercise 7.1.E.1

Prove Theorem 2 in detail.

Exercise 7.1.E.2

Fill in the details in the proof of Corollary 1.

Exercise 7.1.E.2

Prove Corollary 2.

Exercise 7.1.E.3

Show that, in the definition of a semiring, the condition C is equivalent to C.
 [Hint: Consider =AA=mi=1Ai(A,AiC) to get =AiC.]

Exercise 7.1.E.4

Given a set S, show that the following are semirings or rings.
(a) C={ all subsets of S};
(b) C={ all finite subsets of S};
(c) C={};
(d) C={ and all singletons in S}.
Disprove it for C={ and all twopoint sets in S},S={1,2,3,}.
In (a)(c), show that Cs=C. Disprove it for (d).

Exercise 7.1.E.5

Show that the cubes in En(n>1) do not form a semiring.

Exercise 7.1.E.6

Using Corollary 2 and the definition thereafter, show that volume is additive for C -simple sets. That is,
 if A=mi=1Ai(disjoint) then vA=mi=1vAi(A,AiCs).

Exercise 7.1.E.7

Prove the lemma for C-simple sets.
 [Hint: Use Problem 6 and argue as before. ]

Exercise 7.1.E.8

Prove that if C is a semiring, then Cs(C -simple sets )=Cs, the family of all finite unions of C -sets (disjoint or not).
 [Hint: Use Theorem 2.]


7.1.E: Problems on Intervals and Semirings is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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