Question

Please answer both qeustions 1 & 2 with all work. Thank you!

1. Find all solutions of 26 = -8. 2. Let (S. *) and (S', *) be isomorphic binary structures and suppose o an isomorphism from (S, ) to (S', *). Show that is an isomorphism from S' to S. (You may use the fact that the inverse of a bijection is a bijection.)

Answer 1

or, z=(-3t = 1835 eu Lagt fertur)i} for k=0, 12 .5 85 e trzeni 6 l [eos **24T i sin(Tinner a pot cas (1 *2K) A Jin i sin (asayaj for 150, 1,2,. --,5 2= 3t [cos et isln ] for x=0,

for kal z = 35 [ios (1+2) = sin (182) ] = gélius 34+ isin 37] I [ws 7 + is in # ] 2.87 Lo + 1.3 for x=2 2=8* [ews (1+52. 2) a + i sin 2 wor] = gt [les sortisin sof] - 8t (cos (1T-F) is in for-47)] Esto [les In sin ] 2 st pent =st (-va de for x =3, z = 8 trios a tisin Er] 285 [Los(n+7) + isin (1T+1)] =.8t [_cos II --isina] 8 Ev - ]

for k=4 z= 8 (ces an aisin 87). .=8t (cos 34 is in 37) = 8t [cos (57) + is in (rese] = 8t [cost - sin th ] = 8t (6- ) =- ste too far was z=penalan Disin (unuty] = et [ 6013 (27-17+ isin futur = 83 [cos A - isina. = 85 W È] There for all solutions for at g are st (1544), sti, st (Viquin - st(3-7), -sti, st (133)

(2) het a: (s.*) (S, *) be an isomorphism. Then a és a bijective mappeng hence o is invertible ired t exist Also on es bijection, het a, b es het as (a!) = a and Q7 (6)= b Then ' q (a)za' and q (6) = 6 qt (a'xb')= Q+ [9 (2) #' $(6)] pe caca*b)] . since I is a homomorphism a tb since QTQ is the identity mapping ons 3 pula!) * 94660 This proves that ari is a homomorphism since or is a bijection, or is an isomorphism This completes the proof.