14-rec-parcial2.pdf - Escola Polit` ecnica Superior Grau en...

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Escola Polit` ecnica Superior Universitat de Lleida Grau en Enginyeria Inform` atica ` Algebra Febrer de 2014 Recuperaci´ o Parcial 2 Problema 1 Sigui ( G, * ) un grup. Sigui f : A -→ G una aplicaci´ o bijectiva. Aleshores definim una operaci´ o en el conjunt A , de la forma seg¨uent: a b = f - 1 ( f ( a ) * f ( b )) , a, b A. i) Determineu quan l’operaci´o ´ es commutativa. ii) Sigui e el neutre de A per l’operaci´ o . Determineu qui ´ es f ( e ). Considereu ara el cas particular en que G ´ es el conjunt dels enters amb l’operaci´o suma, A el conjunt dels enters senars, i f ( a ) = a - 1 2 . Aleshores, responeu les q¨uestions seg¨uents: iii) Determineu l’aplicaci´ o inversa de f i doneu f - 1 (3). iv) Calculeu 11 ( - 7). v) Doneu una expressi´ o directa per calcular a b a partir dels valors de a i de b . Utilitzeu-la per comprovar la correctesa de l’apartat anterior.
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