15feb-rec-parcial1.pdf - Escola Polit` ecnica Superior Grau...

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Escola Polit` ecnica Superior Universitat de Lleida Grau en Enginyeria Inform` atica ` Algebra Febrer de 2015 Recuperaci´ o Parcial 1 Problema 1 Considereu el conjunt A = { ( a 1 , a 2 , a 3 , a 4 ) : a j ∈ { 0 , 1 }} , on definim la relaci´ o d’ordre R seg¨uent: ( a 1 , a 2 , a 3 , a 4 ) R ( b 1 , b 2 , b 3 , b 4 ) ⇐⇒ ( a 1 , a 2 , a 3 , a 4 ) = ( b 1 , b 2 , b 3 , b 4 ) o b´ e 4 j =1 a j < 4 j =1 b j i) Demostreu que R ´ es antisim` etrica. ii) Determineu si es tracta d’una relaci´ o d’ordre total o d’ordre parcial. iii) Determineu el conjunt B = { ( a 1 , a 2 , a 3 , a 4 ) A : ( a 1 , a 2 , a 3 , a 4 ) R (1 , 0 , 1 , 0) } . iv) Considereu el conjunt S = { (1 , 0 , 0 , 0) , (0 , 1 , 0 , 0) , (0 , 0 , 1 , 0) , (1 , 0 , 1 , 0) } . Doneu el diagrama de Hasse de S i determineu, si existeixen, les cotes inferiors i superiors,
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