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

[Khichdi lover]

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 .
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*

 

[micromass]

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].

 

[Khichdi lover]

Oops. Thanks.

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



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