Question

Find the identity element of z if operation *, defined by a*b = a + b + 1

Answer 1

Answer:

First, we must be dealing with R≠0 (non-zero reals) since 0∗b and 0∗a are not defined (for all a,b).

Let a∈R≠0. An identity is an element, call it e∈R≠0, such that e∗a=a and a∗e=a. Since this operation is commutative (i.e. a∗b=b∗a), we have a single equality to consider.

Suppose we do have an identity e∈R≠0. Then a=e∗a=a∗e=a/e+e/a for all a∈R≠0. This implies that a=a2+e2ae. Thus a2e=a2+e2 and so a2(e−1)=e2 and finally a=±e2e−1−−−√. This is non-sense since a can be any non-zero real and e is some fixed non-zero real. Therefore, no identity can exist.

You may want to try to put together a more concrete proof yourself. Consider for example, a=1. If there is an identity (for a), what would it need to be? Is this possible?