Uni Stuff > Mathematics

Analysis

<< < (2/9) > >>

/0:
Oh, so only 'in general' it's not onto?

Thanks humph :)

/0:
Are the cardinality of all uncountable sets the same? For example, is the set of irrational numbers equivalent to the set of real numbers?

kamil9876:
No, for example there is a famous result that for any set the power set of has strictly greater cardinality. The power set of S is the set of all subsets of S.

therefore

As for the set of irrational numbers, it's cardinality is the same as the cardinality of since is the union of the irrationals and the rationals(rationals are countable) and we have another interesting result you should prove yourself:

if is an infinite set and is a countable set then .

Let be the irrationals, be the rationals.


edit: Reminds me of a very difficult problem you shouldn't waste your time on "Is there a set with cardinality strictly in between the cardinality of integers and the cardinality of reals?".

/0:
Thanks kamil.
I'm not completely familiar with the methods of proof yet so I'll try to prove that a bit later.

Also

--- Quote from: kamil9876 on March 06, 2010, 11:38:50 am ---edit: Reminds me of a very difficult problem you shouldn't waste your time on "Is there a set with cardinality strictly in between the cardinality of integers and the cardinality of reals?".

--- End quote ---

Isn't this the theorem that Cantor spent years on and went insane trying to prove? haha i think i'll pass

/0:
4. How does Zorn's lemma work?
"A partially ordered set in which any chain has an upper bound has a maximal element."

Say you have: , then it doesn't have a maximal element. (or do partial orders require countable sets?)



Thanks :)

Navigation

[0] Message Index

[#] Next page

[*] Previous page