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# 7.1.E: Problemas en Intervalos y Semirings

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## Ejercicio$$\PageIndex{1}$$

Completar la prueba del Teorema 1 y la Nota 1.

## Ejercicio$$\PageIndex{1'}$$

Demostrar Teorema 2 en detalle.

## Ejercicio$$\PageIndex{2}$$

Rellene los datos en el comprobante de Corolario 1.

## Ejercicio$$\PageIndex{2'}$$

Demostrar Corolario 2.

## Ejercicio$$\PageIndex{3}$$

Demostrar que, en la definición de un semiring, la condición$$\emptyset \in \mathcal{C}$$ es equivalente a$$\mathcal{C} \neq \emptyset$$.
$$\left.\text { [Hint: Consider } \emptyset=A-A=\cup_{i=1}^{m} A_{i}\left(A, A_{i} \in \mathcal{C}\right) \text { to get } \emptyset=A_{i} \in \mathcal{C} .\right]$$

## Ejercicio$$\PageIndex{4}$$

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$$(\mathrm{d})$$.

## Ejercicio$$\PageIndex{5}$$

Demuestre que los cubos en$$E^{n}(n>1)$$ no forman un semiring.

## Ejercicio$$\PageIndex{6}$$

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).
\]

## Ejercicio$$\PageIndex{7}$$

Demostrar el lema para$$\mathcal{C}$$ -conjuntos simples.
$$\text { [Hint: Use Problem } 6 \text { and argue as before. }]$$

## Ejercicio$$\PageIndex{8}$$

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

7.1.E: Problemas en Intervalos y Semirings is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.