Auxiliar_4.pdf - Facultad de Ciencias F\u00b4\u0131sicas y Matem\u00b4aticas Universidad de Chile \u00b4 MA1101 Introducci\u00b4 on al Algebra Semestre 2012-02 Profesor

# Auxiliar_4.pdf - Facultad de Ciencias F´ısicas y...

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Facultad de Ciencias F´ ısicas y Matem´aticas Universidad de Chile MA1101 - Introducci´on al ´ Algebra Semestre 2012-02 Profesor: Iv´ an Rapaport Auxiliares: Sebasti´ an Espinosa T., Gonzalo Flores G. Clase Auxiliar N 4: Relaciones e Inducci´ on 30/08 P1. Sea R una relaci´on definida de Z × N \{ 0 } en Z × N \{ 0 } por ( a, b ) R ( c, d ) ad = bc (a) Demuestre que R es relaci´on de equivalencia y determine [(0 , 2)] R es decir, la clase de equivalencia de (0 , 2) Z × N \{ 0 } (b) En Z × N \{ 0 } / R (conjunto cuociente) se define la relaci´ on Ω por [( a, b )] R Ω[( c, d )] R ad bc . Demuestre que Ω es relaci´on de orden y determine si es un orden total o parcial. P2. Sea R la relaci´on en Z × Z definida por ( a, b ) R ( c, d ) a + b 2 c + 3 d (a) Pruebe que R es relaci´on de equivalencia. (b) Muestre que [(0

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