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#1 2011-01-26 10:28:29

euclidcries
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Proving set bijections

Hi, I know about cantor diagonalization argument, but are there any other ways of showing that there is a bijection between two sets? Like, maybe an example using rationals and integers? Or maybe a case where cantors diagonalization argument won't work?

#2 2011-01-26 13:09:16

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Proving set bijections

Hi;

Bijective simply means one to one and onto ( one to one correspondence ). The pickle diagram below shows that the two sets are in one to one correspondence.

200px-Bijection.svg.png


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#3 2011-01-26 14:37:04

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: Proving set bijections

Are you wanting to show that two sets have the same cardinality?


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#4 2011-01-26 16:16:28

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: Proving set bijections

If there exists an injective function from A to B and there exists an injective function from B to A, then |A| = |B|, and therefore there is a bijection between A and B.

Last edited by All_Is_Number (2011-01-26 16:22:44)


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#5 2011-01-27 00:44:01

euclidcries
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Re: Proving set bijections

Hi all,

Bobbym I understand what you are saying, but how do I do this kind of pairing for infinite sets?

Are you wanting to show that two sets have the same cardinality?

Yes...but the only way I know is using Cantor's diagonalization argument

If there exists an injective function from A to B and there exists an injective function from B to A, then |A| = |B|, and therefore there is a bijection between A and B.

Hmm...okay,

when you say "injective function" are you saying one-to-one correspondence? Because that makes sense, if there is a "matching" from A to B and a matching from B to A then there is a bijection.

but, what does the notation |A| or |B| mean? (when you wrote |A| = |B|, what is this statement saying?) Also, how is this done for infinite sets?

Thanks for your replies!

#6 2011-01-27 02:50:24

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

Re: Proving set bijections

euclidcries wrote:

when you say "injective function" are you saying one-to-one correspondence?

No, not quite. An injective function is another name for a one to one function, which implies the following:

if f(a) = f(b), then a = b.

A one to one correspondence is another name for bijection between sets.

but, what does the notation |A| or |B| mean? (when you wrote |A| = |B|, what is this statement saying?) Also, how is this done for infinite sets?

|A| is the cardinality of set A.

By definition, |A| ≤ |B| if there is a one to one mapping of A into B. By the Schröder-Bernstein Theorem, if |A|≤|B| and |B|≤|A|, then |A|=|B|, which implies that there is a bijection between A and B. (See: <http://plato.stanford.edu/entries/set-theory/primer.html#5> and <http://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html>)

I'll try to address the rest of your post later. I have to get to class right now.

Last edited by All_Is_Number (2011-01-27 06:46:52)


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#7 2011-01-27 07:05:48

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

Re: Proving set bijections

euclidcries wrote:

Also, how is this done for infinite sets?

Just because two sets are infinite does not imply that their cardinality is the same. In other words, given two infinite sets |A| and |B|, we cannot conclude that |A|=|B| (although sometimes this is true).

For example,

E = {x | x/2 ∈ ℤ} is the even integers.

let f:E⇾ℤ defined by f(x)=x/2. f is a bijective function, so

|E|=|ℤ|.

But, can you come up with a bijection between ℤ and ℝ?

Last edited by All_Is_Number (2011-01-27 07:06:01)


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