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2.2: Ejercicios, Parte I

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    116013
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    1. Para cada una de las siguientes, escriba Y si la “operación” dada es una operación binaria bien definida en el conjunto dado; de lo contrario, escriba N. En cada caso en que no sea una operación binaria bien definida en el conjunto, proporcione una breve explicación. No es necesario probar ni explicar nada en los casos en los que se trata de una operación binaria.
    1. \(+\)en\(\mathbb{C}^*\)
    2. \(*\)en\(\mathbb{R}^+\) definido por\(x*y=\log_x y\)
    3. \(*\)en\(\mathbb{M}_2(\mathbb{R})\) definido por\(A*B=AB^{-1}\)
    4. \(*\)en\(\mathbb{Q}^*\) definido por\(z*w=\dfrac{z}{w}\)
    1. Definir\(*\)\(\mathbb{Q}\) por\(p*q=pq+1\text{.}\) Demostrar o desacreditar que\(*\) es (a) conmutativo; (b) asociativo.
    1. Demostrar que la multiplicación matricial no es conmutativa en\(\mathbb{M}_2(\mathbb{R})\text{.}\)

    4. Demostrar o desacreditar cada una de las siguientes declaraciones.

    1. El conjunto\(2\mathbb{Z}=\{2x\,:\,x\in \mathbb{Z}\}\) se cierra bajo adición en\(\mathbb{Z}\text{.}\)
    2. El conjunto\(S=\{1,2,3\}\) se cierra bajo multiplicación en\(\mathbb{R}\text{.}\)
    3. El conjunto

    \ begin {ecuation*} U=\ left\ {\ begin {bmatrix} a & b\\ 0 &c\ end {bmatrix}\,:\, a, b, c\ in\ mathbb {R}\ derecha\}\ end {ecuación*}

    se cierra bajo multiplicación en\(\mathbb{M}_2(\mathbb{R})\text{.}\) (Recordemos que\(U\) es el conjunto de matrices triangulares superiores en\(\mathbb{M}_2(\mathbb{R})\text{.}\))

    1. Dejar\(*\) ser una operación binaria asociativa y conmutativa en un conjunto\(u\in S\) Se dice que\(S\text{.}\) un elemento es un idempotente en\(S\) si\(u*u=u\text{.}\) Let\(H\) be el conjunto de todos los idempotentes en\(S\text{.}\) Prove que\(H\) se cierra bajo\(*\text{.}\)
    clipboard_e4119b59eb687081ad91a6a72080e218c.png
    Figura\(\PageIndex{1}\): (Derechos de autor; Bill Griffith. Reimpreso con permiso)

     


    This page titled 2.2: Ejercicios, Parte I is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jessica K. Sklar via source content that was edited to the style and standards of the LibreTexts platform.