Learning Nonstandard Analysis: Understanding Equivalence Relations

[honestrosewater]

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 ))

"<" 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

 

[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]

Define SURE ;)

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 their 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 )