Math Is Fun Forum

dchilow

Member

Registered: 2007-03-05

Posts: 27

Abstract Math Help Please!

Define a sequence of sets E sub n for n≥0 by setting E sub 0 = empty set, and for n greater than 0, E sub n = {E sub n-1}.  Prove that these sets are all distinct, that is, that if m≠n, E sub m ≠ E sub n.

I hope you know what I mean by E sub n, because I don't know how to type this here without MathType.  Please someone help me at least get started because I am confused at what I am supposed to do.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Abstract Math Help Please!

Do you understand what this sequence is?  Write out the first few terms to get a better idea:
[

This is a funky way of writing it, but I believe this is equivalent to Peano's construction of the natural numbers.

Note that E_n has n '{' and n '}' surrounding the null set.  Use strong induction to show that if m != n, then {{...(m times)... { null set } ... (m times) ... }} != {{...(n times)...{ null set } ... (n times) }}.  It seems a bit awkward, but I believe it should work.

Edit: Never mind that.  Simply show that E_n is an element of E_n+1.

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"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..."

dchilow

Member

Registered: 2007-03-05

Posts: 27

Re: Abstract Math Help Please!

Are you saying that the n is the number of '{' surrounding the empty set?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Abstract Math Help Please!

Yep.

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Abstract Math Help Please!

I don’t know if the following proof is okay. I’m not very happy with it myself but I think it might work.

Suppose E[sub]m[/sub] = E[sub]n[/sub] but m ≠ n, say m > n.

But as m − n > 0, E[sub]m−n[/sub] should not be empty. This contradiction means we must have m = n.

dchilow

Member

Registered: 2007-03-05

Posts: 27

Re: Abstract Math Help Please!

Thank you Ricky and Jane.  You were both very helpful.