7.1.E: Problemas en Intervalos y Semirings
- Page ID
- 114039
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Completar la prueba del Teorema 1 y la Nota 1.
Demostrar Teorema 2 en detalle.
Rellene los datos en el comprobante de Corolario 1.
Demostrar Corolario 2.
Demostrar que, en la definición de un semiring, la condición\(\emptyset \in \mathcal{C}\) es equivalente a\(\mathcal{C} \neq \emptyset\).
\(\left.\text { [Hint: Consider } \emptyset=A-A=\cup_{i=1}^{m} A_{i}\left(A, A_{i} \in \mathcal{C}\right) \text { to get } \emptyset=A_{i} \in \mathcal{C} .\right]\)
Dado un conjunto\(S,\) muestran que los siguientes son semirings o anillos.
a)\(\mathcal{C}=\{\text { all subsets of } S\}\);
b)\(\mathcal{C}=\{\text { all finite subsets of } S\}\);
c)\(\mathcal{C}=\{\emptyset\}\);
d)\(\mathcal{C}=\{\emptyset \text { and all singletons in } S\}\).
Desmentirlo por\(\mathcal{C}=\{\emptyset \text { and all } t w o-p o i n t \text { sets in } S\}, S=\{1,2,3, \ldots\}\).
En\((a)-(c),\) espectáculo que lo\(\mathcal{C}_{s}^{\prime}=\mathcal{C} .\) desacredita por\((\mathrm{d})\).
Demuestre que los cubos en\(E^{n}(n>1)\) no forman un semiring.
Usando el Corolario 2 y la definición posterior, muestran que el volumen es aditivo para conjuntos\(\mathcal{C}\) simples. Es decir,
\ [
\ 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}\ derecha).
\]
Demostrar el lema para\(\mathcal{C}\) -conjuntos simples.
\(\text { [Hint: Use Problem } 6 \text { and argue as before. }]\)
Demostrar que si\(\mathcal{C}\) es un semiring, entonces\(\mathcal{C}_{s}^{\prime}(\mathcal{C} \text { -simple sets })=\mathcal{C}_{s},\) la familia de todas las uniones finitas de\(\mathcal{C}\) -conjuntos (disjuntas o no).
\(\text { [Hint: Use Theorem } 2 .]\)