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

http://matematica.noads.biz

http://matematica.noads.biz

http://matematica.noads.biz

_1352045656.unknown

_1352046245.unknown

_1352047422.unknown

_1352121808.unknown

_1352122872.unknown

_1352122913.unknown

_1352122181.unknown

_1352122274.unknown

_1352121835.unknown

_1352121754.unknown

_1352121795.unknown

_1352047448.unknown

_1352047036.unknown

_1352047200.unknown

_1352047334.unknown

_1352047181.unknown

_1352046464.unknown

_1352046728.unknown

_1352046398.unknown

_1352045918.unknown

_1352045990.unknown

_1352046204.unknown

_1352045776.unknown

_1352045813.unknown

_1352045870.unknown

_1352045675.unknown

_1351523512.unknown

_1351524087.unknown

_1351524181.unknown

_1351524216.unknown

_1351524319.unknown

_1352045614.unknown

_1351524441.unknown

_1351524280.unknown

_1351524154.unknown

_1351523898.unknown

_1351523978.unknown

_1351524022.unknown

_1351523935.unknown

_1351523715.unknown

_1351523820.unknown

_1351523677.unknown

_1335619030.unknown

_1335619818.unknown

_1337067399.unknown

_1337067581.unknown

_1351523447.unknown

_1337067550.unknown

_1335619924.unknown

_1335620403.unknown

_1335620465.unknown

_1337067292.unknown

_1335620440.unknown

_1335620377.unknown

_1335619886.unknown

_1335619403.unknown

_1335619723.unknown

_1335619761.unknown

_1335619478.unknown

_1335619249.unknown

_1335619370.unknown

_1335619088.unknown

_1335616099.unknown

_1335618672.unknown

_1335618980.unknown

_1335618611.unknown

_1335458059.unknown

_1335615939.unknown

_1335457916.unknown