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        riddles >> putnam exam (pure math) >> Symmetric Difference Group
        
(Message started by: Icarus on Dec 8^th, 2005, 2:35pm)

Title: Symmetric Difference Group
Post by Icarus on Dec 8^th, 2005, 2:35pm

The symmetric difference between two sets A and B is the set
A v B = (A u B) - (A n B) = (A - B) u (B - A).

(1) For any set S, show that (P(S), v) is a group, where P(S) is the set of all subsets of S.
(2) Identify all subgroups of P(S) when S is finite.
(3) What if S is infinite?

Title: Re: Symmetric Difference Group
Post by Eigenray on Dec 9^th, 2005, 12:29pm

I'd like to add (multiply?):
(4) Show that (P(S), v, n) is a ring, where addition is set difference v, and multiplication is set intersection n.
(5) Identify (and enumerate) all subrings of P(S) when S is finite.
(6) What if S is infinite?

(When I first tried solving (2), a bell went off in my head, and I realized I had done (5), which I found to be a sterling problem.)