R.D Silverman's number theory homework

[Mr. P-1]

Originally Posted by davar55

Infinity ... is not a "Real Number" as in the closure of the rational numbers.

The real numbers are the completion of the rational numbers. Closure is a different concept.

[science_man_88]

Originally Posted by Mr. P-1

1)Try to complete this sentence:

A partition on S is a collection of ______ of S, such that ...



OK for now.



Perfect.



A collection is a set of sets. That means that everything you know about sets applies to collections too.

The word "group" has a technical meaning in mathematics and really shouldn't be used, even informally, to mean anything else.

1) Pairwise disjoint subsets of S, would be my best guess.

dearly noted on the word group I just didn't see set as a possibility but I should have though of the fact that the sets are subsets of the collection.

[science_man_88]

Originally Posted by Mr. P-1

An intersection between two sets is the set whose elements are just the elements that the two sets have in common.

We haven't talked about it yet, but how do you think you could define the intersection of a collection of sets?



Complete the following sentence:

The union of two sets is the set whose elements are...the distinct elements of A and B such that no overlap occurs.

Try to define the union of a collection of sets.
the collection itself with all duplicates thrown away ?



Formally defining an n-tuple require concepts more advanced than we've discussed. Informally, an n-tuple is an ordered set of n not necessarily distinct elements.



Complete the following sentence:

A Cartesian product A x B of two sets is the set of all ordered pairs(a, b) with aA and b in B

(Hard) How would you define a Cartesian product on an n-tuple of sets?



What's a relation? What kind of things?

good god I'm screwed I can't figure a way to answer either , especially describing a relation without using the word relation.

Perfect (assuming you can define reflexive symmetric and transitive).



As I said, I think you're making great progress.

here's what you wanted as feedback, assuming feedback isn't negative.

[Chris Card]

Originally Posted by science_man_88

What's a relation? What kind of things?

good god I'm screwed I can't figure a way to answer either , especially describing a relation without using the word relation.

Something like "a relation between two sets A and B is a subset of the Cartesian product A x B"?

Chris

[xilman]

Originally Posted by Mr. P-1

Not to me. At least, if he has, then I'd be concerned about the quality of the source. Although a considerable improvement upon his previous efforts, his definitions still lack precision, leaving me uncertain whether he doesn't properly understand the concepts, or he does, but just can't explain them precisely. More and more I find myself leaning toward the latter, which is a clear indication of progress.

Ok, I'm quite prepared to admit that I may be wrong in this instance. If so, I apologise to sm88 for my statement.

Paul

[science_man_88]

Originally Posted by xilman

Ok, I'm quite prepared to admit that I may be wrong in this instance. If so, I apologise to sm88 for my statement.

Paul

paul I like memorization so I should be able to memorize book like definitions so I don't blame you.

[davar55]

Originally Posted by Mr. P-1

The real numbers are the completion of the rational numbers. Closure is a different concept.

@P-1: For me at least please define both as you see the difference.

[science_man_88]

Originally Posted by davar55

@P-1: For me at least please define both as you see the difference.

http://en.wikipedia.org/wiki/Closure_(mathematics)

I can't find one for completion

[CRGreathouse]

Originally Posted by science_man_88

I can't find one for completion

http://en.wikipedia.org/wiki/Completeness_(order_theory)

See also
http://terrytao.wordpress.com/2010/1...dard-analysis/

[Mr. P-1]

Originally Posted by CRGreathouse

http://en.wikipedia.org/wiki/Completeness_(order_theory)

See also
http://terrytao.wordpress.com/2010/1...dard-analysis/

And, for the sake of, er, completion, this:

http://en.wikipedia.org/wiki/Complet...ace#Completion

[R.D. Silverman]

Originally Posted by Mr. P-1

And, for the sake of, er, completion, this:

http://en.wikipedia.org/wiki/Complet...ace#Completion

The intermediate value theorem strikes again!