You need to exhibit the identity. What does the identity do, in this case? Well, given any set you have that right? Now, and must be the same kind of object; that is, members of Which member of will do what you need?
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.
So, Plato has told you that the identity is the empty set. You now need to prove this this identity is unique, and you need to find the inverse of an element.
To prove the identity is unique you do not need to use this example - any identity of a set is unique. This is because if you have two identities and then clearly but also . Thus they are equal.
To find the inverse, you want to find a set such that . Note that and . What does this tell you? (Remember you want ).