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#1 2006-04-24 13:35:51



Proofs about Sets

I need to prove the following:

a) Let A, B be sets contained in a universal set U. Suppose that A ≈B. prove that P(A) ≈ P(B) (power sets)
b) Let A, B be sets contained in a universal set U such that A is a subset of B. Suppose that A is countable and B is uncountable. prove that B - A is uncountable.
c) Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent.

Shroeder-Bernstein Theorem: Let A and B be sets, and suppose that A <= B and B <= A. Then A ≈ B.

Can anyone help with any of those?

#2 2006-04-25 00:59:32



Re: Proofs about Sets


Since A ~ B, there exists a 1-1 and onto function θ.  So consider all the power sets of A.  Just take every a in there, and replace it with θ(a).  All these elements must be distinct, so as long as the power sets of A are distinct (which they are), then these will be the power sets of B.


Taking a countable number of elements away from an uncountable set, it should be clear this will leave you with an uncountable set.

Let's assume the B - A is countable.  Then B - A = {x : x∈B and x is not in A}.  Consider B - A + (B intersect A) = B.  B intersect A is countable since A is countable.  Thus, we have B - A + (B intersect A) which must be countable since both B-A and B intersect A are countable.  So B is countable.  Contradiction.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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