What is the identity element in abstract algebra groups?

In summary, the conversation discusses the symmetric difference operation, which is denoted as A + B = (A - B) U (B - A). The identity element, denoted as e, is also mentioned, and it is stated that for A + e = A to be true, e must be the empty set. The second case, (e - A) = A, is also discussed and it is mentioned that this statement cannot be undefined for the conditions of a group to be met. The concept of a set X is also brought up in relation to the equation.
  • #1
IKonquer
47
0
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?
 

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  • #2
If A=A'(inverse) then why does A+A'={}(empty set)?
 
  • #3
A + A' is the symmetric difference, and not by means of normal addition.
 
  • #4
Ah. Well I learned something :)
 
  • #5
(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?
 
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1. What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It is a more generalized and abstract approach to algebra, focusing on the properties and relationships of these structures rather than specific numerical computations.

2. What is a group in abstract algebra?

In abstract algebra, a group is a set of elements with a defined binary operation (usually denoted as *) that satisfies four main properties: closure, associativity, identity, and invertibility. These properties allow for the manipulation and study of abstract structures without necessarily relying on numerical values.

3. What are some examples of groups in abstract algebra?

Some examples of groups in abstract algebra include the integers under addition, the rational numbers under multiplication, and the set of all invertible 2x2 matrices under matrix multiplication. Other common examples include symmetric groups, dihedral groups, and cyclic groups.

4. How is abstract algebra used in other branches of science?

Abstract algebra has many applications in various branches of science, including physics, computer science, and cryptography. It is used to study and model symmetries in physical systems, design efficient algorithms for computations, and create secure encryption methods.

5. What are some open problems in abstract algebra?

Some open problems in abstract algebra include the classification of finite simple groups, the existence of certain types of non-associative algebras, and the study of infinite-dimensional groups and their representations. These problems are actively being researched and have implications in various fields of mathematics and science.

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