Here is another part of the question:
For part (c), this is what I've done so far:
suppose both e and é are identities of and suppose . Therefore
Since , we can set x=é in equation (1) and set x=e in equation (2)
Thus e and é are equal and the identity is uniqe.
Is this correct? I think the problem here is that I've shown that the identity of is unique but I have not shown that it must have an identity. Can anyone help?
For part (d), I did this:
Suppose b and c are both inverses of a. Then also . Hence .
Again, I think my approach just shows the uniqueness, and not the existence of an inverse... I appreciate any help with this.