Exerciii rezolvate cu legi de compoziieEnunuri Ex.1.Pe mulimea numerelor reale definim operaia , oricare ar fi .a)Verificai identitatea , oricare ar fi .
b)Demonstrai c , oricare ar fi .
c)Artai c legea de compoziie este asociativ.
d)Calculai .e)Rezolvai in ecuaia .
Variante M2 bac 2009Ex.2.
Variante M2 bac 2009Ex.3.
Variante M2 bac 2009Ex.4.
Variante M2 bac 2009Ex.5.Pentru a,b din mulimea se definete operaia .
Variante M1 bac 2009
Rezolvri:
Ex.1.
a) i identitatea din cerin este demonstrat.b).c)Legea de compoziie este asociativ dac .
Din cele dou relaii de mai sus rezult c legea este asociativ.
d) conform punctului b).e)
Ecuaia dat devine .
Cum rezult c .Ex.2.
2.a) c.c.t.d.
b) care are soluiile i .
c)Observm c
.
Ex.3.
2.a) c.c.t.d.
b).
c)S mai observm c i .
Observai in acea compunere rolul important al lui 2!Utilizand proprietatea de asociativitate a operaiei precum i faptul c i se
obine c E=2.
.Ex.4.
a)e este element neutru dac .
b)
Ecuaia dat devine:
(am folosit formula .
Obinem ecuaia care are soluiile .
c).
Observm c dac lum i vom obine
Din obinem iar din obinem .
Evident a i b sunt numere raionale .Ex.5.a)Fie
b)Legea de compoziie * este asociativ dac deci legea este asociativ.
c)
Demonstrm prin inducie c este adevrat.Etapa verificrii:
Pentru n=1 avem este adevrat.
Etapa demonstraiei:
Presupunem P(k) adevrat i demonstrm c P(k+1) este adevrat.
este adevrat.
trebuie demonstrat.
c.c.t.d.
Egalitatea devine
.Notm i obinem ecuaia de gradul doi
care are soluiile i .
Revenind la notaie , obinem sau
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