question_answer 53)

Show that none of the operations given in example 9 (except part (v)) has identity. 

Answer:

Let ?e? be the identity element in Q.       (i)         a ? e = e ? a = a       a ? e = and e ? a = a       e = 0 and e =       which is not possible for every Hence there is no identity element.       (ii)   a * e = e * a =        a2 + e2 = e2 + a2 = a                   Let a = 3              there is not identity element. (iii)  a * e = e * a =                                            is an identity element. (vi)  a  e = e* a =                               which is not possible.        Identity element does not exist.