7.1.E: Problems on Intervals and Semirings
( \newcommand{\kernel}{\mathrm{null}\,}\)
Complete the proof of Theorem 1 and Note 1.
Prove Theorem 2 in detail.
Fill in the details in the proof of Corollary 1.
Prove Corollary 2.
Show that, in the definition of a semiring, the condition ∅∈C is equivalent to C≠∅.
[Hint: Consider ∅=A−A=∪mi=1Ai(A,Ai∈C) to get ∅=Ai∈C.]
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 two−point sets in S},S={1,2,3,…}.
In (a)−(c), show that C′s=C. Disprove it for (d).
Show that the cubes in En(n>1) do not form a semiring.
Using Corollary 2 and the definition thereafter, show that volume is additive for C -simple sets. That is,
if A=m⋃i=1Ai(disjoint) then vA=m∑i=1vAi(A,Ai∈C′s).
Prove the lemma for C-simple sets.
[Hint: Use Problem 6 and argue as before. ]
Prove that if C is a semiring, then C′s(C -simple sets )=Cs, the family of all finite unions of C -sets (disjoint or not).
[Hint: Use Theorem 2.]