Let I be any set and let R be the collection of all subsets of I. Define addition and multiplication of subsets A,B ⊆ I as follows:
A+B = (A∪B) ∩ [A∩B(with a line over this)] and AB = A∩B. Show that R is a commutative ring under this addition and multiplication.
I know that to be a commutative ring it has to have the following:
closure, associativity, commutativity, distribution, identity and inverses.
Closure - show well-defined
Assoc - (a+b)+c = a+(b+c) and a(bc) = (ab)c
Comm - a+b = b+a and ab = ba
Dist - a(b+c) = ab+ac and (a+b)c = ac+bc
I am not sure how to write them out using the defined subsets above.