Question

Which of the following sets, together with the given binary operation *, DOES NOT form a group? (Notation: As usual, the notations Z, Q, R, and C represent the sets of integers, rational numbers, real numbers, and complex numbers, respectively.) (A.) G is {a+bV2 ER\{0} | a, b e Q}; * is the usual multiplication of real numbers (B.) G is {a + biv2 € C\{0} | a, b E Q}; * is the usual multiplication of complex numbers (C.) G is {Ta e R|a € Z}; * is given by Ya* = a + b for all ya and (D.) G is R\{0}; * is given by a *b = |a| · b for all a and b in G (E.) G is {z € C||2= 1}; * is the usual multiplication of complex numbers 6 in G

Answer 1

G= } at brz EiRlcoy l a, b c Q} closure : Let oc=ar to Tz, ya ã + brz ein then ny - (at buzz ( a + love) - a ă + 2 btt a b 2 + a 5 ja = (aã+265) + ab + ab) va & G closed under multiplication le ssociative - x, y, z E G Gay, z = recyzy hold 7 e=1tore Identity a= at bisa such that ap = P x = re. Inverse - for a= at b52 7 a-bre en Q²-262 such that xy = y x = e

Page No.: Group (B) honce Giren <G -> forn, G= { at Ebral a, b Eos closure for x,y € 9 ez at i Brz cted ja y xxy = fat i bej (credaj - ac-abdi te (bctad) z E G. Associative & ny, z E G (* y) * 22 x + (y *2 Identing for x= at ibJz ey = 1+ lore x+y = (at e Bje) * (teory = at ibiz 90 y = It iora form Identity for G

Inverse :- for x= aribre 7 x'= a -i buz in G a 2+262 such that xx x's xxney where y is Identity hence <6*> form Group G2 z 31 a. I a Eg} o closure Axiom for to y E G sasa y = 36 sty Tax = sja+b e G Asso că atue Axiom for any z E G se ata yz 25, 2= 7 Flatbtc 4*4* ,= xt Cy*2 =

Identity : por x = Ja & G 7 e=0EG xoch that v x= a = x e* - 2 * e = Inverse - for or= 3 a E G 7 y = fa then => (0) ** y = y + x = 0 Ç from Group G={ x ElRusoy} closure Axiom for a, y eG xxy - 11loy & G A Asociative Axiom For oc, y, ze 9 (2*2 *2= 1 BC*yl.2 = 1 121-yl z = 1 @ I lylz

hence x*(y + 2) = x* (1y1-2) = 1x) - ly 1 - 2 bence x* (y*) = (x+y) * ¥ x, y, z eng Identity. 7 Any Identing e & 9 such that sete fe** <G *7 form Group (6 G=zxed 1 1x) = 15 closure Axiom - for x, y E G 1x1=1 & lyl=1 Now lxy) = lal-lyl=1 - - xy & G Aspo ceature Axiom - for x,y,z & G 168 z= x lyzj hold

Page No.: Identity Axion 7e se G such that I ne G е се ж Inverse - for de G 7 g 7 a E 7 Tuch that x-x= xona lapel hence <G -7 ore Group option Doesn't they only form (A) Group