Is There an Error in the Additive Inverse Definition in Rudin's PoMA?

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In summary, Rudin PoMA chapter 1 in appendix 1 gives a definition of the additive inverse of a cut in rational numbers. Part 1 of the proof is that any element of \alpha+\beta should be a negative rational. However, part 2 of the proof is that if r \in \alpha and s \in \beta, then -s \notin \alpha. This error was pointed out in the errata for Rudin's book, but was not in the book itself.
  • #1
Khichdi lover
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In Rudin PoMA , chapter 1
in appendix 1 construction of real numbers as Dedekind Cut's is given.

I feel there is an error in the definition of additive inverse of a 'cut' .

given a cut [itex]\alpha[/itex] in rational numbers , its additive inverse is given by [itex]\beta[/itex] .

a rational p belongs to [itex]\beta[/itex] , if there exists a rational number r>0 such that -p-r[itex]\notin[/itex] [itex]\alpha[/itex] .

the additive identity 0* is the set of all negative rational numbers.

No problem till this point.

Then we are supposed to prove that [itex]\alpha[/itex] + [itex]\beta[/itex] = 0.

For this ,part 1 of the proof is that any element of [itex]\alpha[/itex] + [itex]\beta[/itex] should be a negative rational.Here's Rudin's proof
If r [itex]\in[/itex] [itex]\alpha[/itex] and s [itex]\in[/itex] [itex]\beta[/itex] ,
then -s [itex]\notin[/itex] [itex]\alpha[/itex] , hence r<-s , r+s < 0 . Thus [itex]\alpha[/itex]+[itex]\beta[/itex] [itex]\subset[/itex] 0*

How does -s [itex]\notin[/itex] [itex]\alpha[/itex] follow from s [itex]\in[/itex] [itex]\beta[/itex] ? I feel this is printing error.(do you agree on this?)

Anyway, the proof can be slightly changed to make it correct :-
If r [itex]\in[/itex] [itex]\alpha[/itex] and s [itex]\in[/itex] [itex]\beta[/itex] ,
then there is a rational number t>0 , such that
- s - t [itex]\notin[/itex] [itex]\alpha[/itex],
hence r < -s - t ,
r + s < - t < 0.

Am I right ? I ordered the book's 3rd ed in India, and I am discovering that the book has many errors . :cry:
I checked this errata - http://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_notes.pdf , but couldn't find the error I pointed out.

*note :- if this isn't posted in right sub-forum,Please move this to appropriate forum, in which one is supposed to discuss errata*
 
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  • #2
Rudin is correct. There exist a rational t>0 such that -s-t is not in [itex]\alpha[/itex]. But

[tex]-s-t<-s[/tex]

So -s is also not in [itex]\alpha[/itex].
 
  • #3
Oops. Thanks.

I got confused between [itex]\notin[/itex] and [itex]\in[/itex] .

:blushing:

Doubt resolved. Please lock the thread if the need be.
 

FAQ: Is There an Error in the Additive Inverse Definition in Rudin's PoMA?

What is the purpose of the Rudin Errata?

The Rudin Errata is a document that contains corrections and updates to the textbook "Principles of Mathematical Analysis" by Walter Rudin. It is intended to help readers identify and correct errors in the book.

How can I access the Rudin Errata?

The Rudin Errata can be found online through various sources, including the publisher's website or through a simple internet search. It is also included in newer editions of the textbook.

Who creates the Rudin Errata?

The Rudin Errata is typically created by a team of mathematicians and experts who carefully review the textbook for errors. These individuals may also consult with the author, Walter Rudin, for clarification on certain issues.

Is the Rudin Errata necessary to use the textbook?

While the textbook can be used without consulting the Rudin Errata, it is highly recommended to have access to it in order to have the most accurate and up-to-date information while studying the material.

What should I do if I find an error not listed in the Rudin Errata?

If you discover an error in the textbook that is not listed in the Rudin Errata, you can contact the publisher or the author to report it. They may include the correction in future editions of the textbook.

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