What is the identity element in abstract algebra groups?

[IKonquer]

The .pdf can be ignored.

Let A + B = (A - B) U (B - A) also known as the symmetric difference.

1. Look for the identity and let e be the identity element

A + e = A
(A - e) U (e - A) = A

Now there are two cases:

1. (A - e) = A
This equation can be interpreted as removing from A all elements that belong to e to yield the set A. In order for this statement to be true, the identity element e must be the empty set.

2. (e - A) = A
This equation can be interpreted as removing from e all elements that belong to A to generate a set A. Is this statement undefined?

 

[ArcanaNoir]

If A=A'(inverse) then why does A+A'={}(empty set)?

 

[IKonquer]

A + A' is the symmetric difference, and not by means of normal addition.

 

[ArcanaNoir]

Ah. Well I learned something :)

 

[DizzyGillespie]

(e-A) must equal something else and not A. Moreover it must equal something such that the union of (A-e)=A with (e-A)=X is A U X=A. I am sure you are aware of such a set =).

It can't be undefined or else were breaking the conditions of what it is to be a group. A and e are elements of the group so (A-e)U(e-A) must be too. right?