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Proving that Z2 X Z2 X Z2 . Z2 is a isomorphic (ring isomorphism) to P(N)

  1. Apr 19, 2012 #1
    Proving that Z2 X Z2 X Z2..... Z2 is a isomorphic (ring isomorphism) to P(N)

    1. The problem statement, all variables and given/known data
    I wish to prove that the ring of Cartesian product Z2 X Z2 X Z2....X Z2 (here we have n products) under addition and multiplication (Z2 is {0,1}) is isomorphic to P(N) where P(N) is the ring of power set of the set of n elements where the multiplication operation is AB=A U B and the addition operation is A+B=(A U B)-(A[itex]\cap[/itex]B)


    2. Relevant equations
    Both of these rings are Boolean rings
    A ring isomorphism is an operation preserving (both operations) one to one and onto map between the two rings


    3. The attempt at a solution
    I tried for smaller cases such as Z2 => P(1) and I was able to find an isomorphism through brute force. I want to somehow use the fact that these two rings are Boolean rings to solve this.
    While searching for a solution I ran into something called "the Stone Representation theorem" which states that all Boolean Algebras are isomorphic to a field of sets. Is it possible to apply this theorem in this case?

    Thank you for the help
     
  2. jcsd
  3. Apr 20, 2012 #2
    Re: Proving that Z2 X Z2 X Z2..... Z2 is a isomorphic (ring isomorphism) to P(N)

    Do you know how to prove the isomorphism in the case where the multiplication operation is intersection, rather than union? Think about what happens when each set is replaced with its complement.
     
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