Nonstandard Analysis

[honestrosewater]

:confused:
I'm reading a book and am stuck. Here's an excerpt:

(ByTheWay- this is an introduction to nonstandard analysis and we are defining an equivalence relation which we will use to construct the hyperreals from the rationals via Cauchy sequences (I think :biggrin: ))

"<" and ">" will enclose subscripts
"[" and "]" will enclose the English description of the symbol

BEGINNING OF EXCERPT

"Let r = (r<1>, r<2>, r<3>, ...) and s = (s<1>, s<2>, s<3>, ...) be real-valued sequences. We are going to say that r and s are equivalent if they agree at a "large" number of places, i.e., if their 'agreement set'

E<rs> = {n : r<n> = s<n>}

is large in some sense that is to be determined. Whatever "large" means, there are some properties we will want it to have:

1) N = {1, 2, 3, ...} must be large, in order to ensure that any sequence will be equivalent to itself.

2) Equivalence is to be a transitive relation, so if E<rs> and E<st> are large, then E<rt> must be large. Since E<rs> [intersection] E<st> [proper subset] E<rt>, this suggests the following requirement:

If A and B are large sets, and A [intersection] B [proper subset] C, then C is large.

In particular, this entails that if A and B are large, then so is their intersection A [intersection] B, while if A is large, then so is any of its supersets C [proper superset] A."

END OF EXCERPT

Where I'm stuck: "Since E<rs> [intersection] E<st> [proper subset] E<rt>"
I see that E<rt> can NOT be a proper subset of the intersection of E<rs> and E<st>. I see that the intersection of E<rs> and E<st> CAN be a proper subset of E<rt>, but I can't see why the intersection of E<rs> and E<st> MUST be a proper subset of E<rt>. Why can't they be equal?

(ByTheWayAgain the book is "Lectures on the Hyperreals" by Robert Goldblatt)

Many thanks for any help.

Rachel

I don't have a cool tag line :rolleyes:

[HallsofIvy]

Are you SURE about the "proper" part? Some authors use the "subset" symbol, without the additional line, to mean simply "subset" and not necessarily "proper subset".

[honestrosewater]

He uses the- how to say- horizontal UI which I had always seen defined as proper subset. But this morning I did find it in one other book (out of a dozen!) to mean subset. More importantly, I -duh- looked on the next page where he uses UI to define power sets.
So problem solved. Thanks again.

Rachel

P.S. Just thinking aloud- no need to reply- but if it WAS proper subset, is there any way it could work? Actually, that makes me think of a question.

He says that r and s are real-valued. This only means that their range, or the values of thier terms, is/are a subset of R, not that their range is equal to R?
Sorry, another quick one- when he says
r = (r<1>, r<2>, r<3>, ...)
I should assume the sequence is infinite, since he doesn't name even a general last term? BTW he doesn't make any comments on conventions. (Or include a list of symbols ;) )

Oh, yes and

Happy Easter!

(if applicable :smile: )