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 $\alpha$ in rational numbers , its additive inverse is given by $\beta$ .

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

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

No problem till this point.

Then we are supposed to prove that $\alpha$ + $\beta$ = 0.

For this ,part 1 of the proof is that any element of $\alpha$ + $\beta$ should be a negative rational.Here's Rudin's proof

If r $\in$ $\alpha$ and s $\in$ $\beta$ ,
then -s $\notin$ $\alpha$ , hence r<-s , r+s < 0 . Thus $\alpha$+$\beta$ $\subset$ 0^*

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

Anyway, the proof can be slightly changed to make it correct :-
If r $\in$ $\alpha$ and s $\in$ $\beta$ ,
then there is a rational number t>0 , such that
- s - t $\notin$ $\alpha$,
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 $\alpha$. But

$$-s-t<-s$$

So -s is also not in $\alpha$.

 

[Khichdi lover]

Oops. Thanks.

I got confused between $\notin$ and $\in$ .



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