How to prove that (R,*) is an abelian group.

generally approach this proof:

1. Closure: You need to show that if a and b are any elements in R, then a * b is also an element of R.

2. Associativity: For any a, b, and c in R, you must prove that (a * b) * c = a * (b * c).

3. Identity Element: There should be an element e in R such that for every a in R, a * e = e * a = a.

4. Inverses: For each element a in R, there should be an element b in R such that a * b = b * a = e, where e is the identity element.

5. Commutativity: For every a and b in R, a * b = b * a, which is what distinguishes an abelian group from just a group.

Step by Step Proof Concept: - For closure, you would need to take any two elements from the set R, perform the operation *, and verify that the result is still in R. - For associativity, a practical approach would be to select elements a, b, and c from R and check the outcome of the operation in both orders of pairing to confirm they are the same. - For the identity element, you need to find a specific element in R that acts as a neutral element when * is applied with any other element from R. - To find inverses, for each element in R you must find another element in R that can be combined with the first to yield the identity element of the group. - Finally, for commutativity, you simply have to show that any two elements can be combined in any order and the result will be the same.

You would need to provide concrete examples and arithmetic or algebraic proof for each of the properties if you were working with a specific set R and operation *.