Question

Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just try to determine one group axiom that fails to hold): 1) The set of polynomials of odd degree under addition. 2) The set of polynomials of odd degree under multiplication 3) The set of integers congruent to 1 modulo 11, under addition modulo 11. 4) The set of integers modulo 11, under multiplication modulo 11. 5) The set of nonzero integers modulo 4, under multiplication modulo 4.

Answer 1

To I I the set of polynomials of odd degree under addition a-şaintautasust...} there ai, az, as, ..if R and a , az, as,... are - not all zero i additine G is without identity failed hobe a group Hence it ♡ G = { 0, nagtagust... as ative Ста, ье a = aint az ut.n. b= bint by 13 t non - aba a, b, a+(a, but az 6 ) x4 - Failure of closure Property cs Scanned we can not be a group

{ © C = {x/n= 1 (model) } 10, 2, 1, 12, 23, ...} for 1, 12€ la - 1+12=13 & 2 (modll) Failure of the closure Property G is not a group. 0 G = {2/2=0 (modll) { 0,111, 122, 133, - for at la 7 no be a I ab = a (modll) in Cs donot contain multiplicative - identity, G failed to be a group Scanned with CamScanner

③ G = { xE74-{o} / 2 = 0 (mody)} for 4 ta I no at G эча - ч( Modч) the failure of multiplicatine identity G Can't be a group under cs Scanned with mulliplication. CamScanner