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USTHB-Faculté de Mathématiques1ère • Aiinêe~Liçe&oi& SM^Sëctâon. A;10 janvier .2012Durée J.H30Examen final dePartie 1 AlgèbreExercice 1. (4points)On définit sur l'ensemble R la relation R, définie, par(1) Montrer que 72. est une relation d'équivalence.(2j • Déterminer la classe d'équivalence jdc- x — |Exercice 2. (6 points)Dans l'ensemble E =E - {-1} on considereMN(2) Mbntrer que ialôj(3) MontrerPartie 2-ArialyseExercice 3.On considèT'et (%} données par2t%(i) Montrer par récurrence quç ^ - vn.= :U° .'"0;: n €'(ii) En déduire que.un < vn et '~Bm un -vn.>(iii) Montrer que (un) est croissante st que (vj est décroissante. =(iv) Montrer que les suite (ti,*) et (vn)sônt convergentes et qu'eUësont la ttiêmme limite t.': . .Exercice 4. (4 points)Soit (a, 6) 6 SI2 et / la fonction définie sur R par :,siz. < 0Etudiez, suivant tes vafeiar^de 4 et è làmttUSTHB-Faculté de Mathématiques1ère Année-Licence SM-Section ACorrigé de l'Examen final de Mathl"Cerrigé de l'exerciceSoit 7£ la relation définie sur R parxH-y x - y - je1(1) Montrons que K est une relation d'équivalencece(a) Puisque Vx 6 R on a 0 = = x — x — x'2 — x- alorsqui montre -que cette relation est reflexive(b) Si on a xTiy* ce qui vêt dire que x — y — .T- — y2 alorsy — îc = y2 — -x2 ce. qui montre que yTLx et donc querelation, est symétrique.(c.) Soient x, y, z € M. tels que'x"R,y.etalors on a? ...
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