This tag is for elementary questions on set theory - the sort of material covered in "Chapter 0" of graduate texts and in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, ...
6
votes
2answers
683 views
bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?
Thank you.
33
votes
4answers
2k views
How do I define a bijection between $(0,1)$ and $(0,1]$?
How do I define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if ...
2
votes
2answers
617 views
Countable Sets and the Cartesian Product of them
If I have two countable sets, $A$ and $B$, how can I prove that the cartesian product of them, $A \times B$, is also countable?
18
votes
6answers
2k views
In set theory, how are real numbers represented as sets?
In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?
Thanks for the answers. Several answers ...
15
votes
2answers
464 views
How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$.
What is the proof of this fact?
As Arturo pointed out in his comment, in computer science this ...
8
votes
4answers
566 views
Countable set having uncountably many infinite subsets
Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite ?
15
votes
2answers
762 views
Infinite product of measurable spaces
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product ...
16
votes
1answer
538 views
Characterising functions $f$ that can be written as $f = g \circ g$?
I'd like to characterise the functions that ‘have square roots’ in the function composition sense. That is, can a given function $f$ be written as $f = g \circ g$ (where $\circ$ is function ...
2
votes
2answers
1k views
countably infinite union of countably infinite sets is countable
How do you prove that any collection of sets {$X_n : n \in \mathbb{N}$} such that for every $n \in \mathbb{N}$ the set $X_n$ is equinumerous to the set of natural numbers, then the union of all these ...
10
votes
3answers
461 views
The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is ...
22
votes
6answers
2k views
How does Cantor's diagonal argument work?
I'm having trouble understanding Cantor's diagonal argument. Specifically, I do not understand how it proves that something is "uncountable". My understanding of the argument is that it takes the ...
18
votes
3answers
918 views
difference between class, set , family and collection
In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
18
votes
4answers
562 views
Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of ...
23
votes
3answers
3k views
Proof that the irrational numbers are uncountable
Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the ...
12
votes
3answers
544 views
The Aleph numbers and infinity in calculus.
I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus?
Are they interchangeable/transposable in ...
