Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on NN given by f≡g⟺{n∈N:f(n)=g(n)}∈U, where U is a non-principal ultrafilter on N. The question is to find the cardinality of the set of all equivalence classes. My guess would be 2ℵ0, but, well, it is just a guess. I don’t really know how to tackle this exercise, not even … Read more

If the set of limit points of A⊆RnA \subseteq \mathbb{R}^n is countable, then AA is countable

Let A⊆Rn. Prove that if A′ is countable, then A is countable. A′ is the collection of all limit points of A. This is the problem on Real Analysis. I know how to prove injection on A to N={0,1,2,…}, but I have no idea how to do the above problem. Answer HINT: Prove the contrapositive. … Read more

Showing that |clubκ|>κ |\operatorname{club}_\kappa| > \kappa

I’m trying to solve the following exercise: Show that |clubκ|>κ Worded differently: Show that there are more than κ closed and unbounded subsets of κ I think this might be some sort of standard diagonal argument, however, I have no idea where to begin I would appreciate some help Answer Here is a simple construction. … Read more

Filters invariant under bijections

Which filters on a set A are invariant under every bijection A↔A? I found the following examples of invariant filters: {A∖K∣K∈PA,cardK≤κ}, for every cardinal number κ. {A∖K∣K∈PA,cardK<κ}, for every cardinal number κ. (added) Are there any other examples? By the way, are my two classes of examples equal to each other? Answer Your second type … Read more

Suppose A, B and C are sets such that #A=#B. Is it true that #(A x C) =#(B x C)? If it is, prove that. [duplicate]

This question already has an answer here: Arithmetics of cardinalities: if |A|=|C| and |B|=|D| then |A×B|=|D×C| (1 answer) Closed 5 years ago. I’m sure that this is true, because A and B have the same cardinality (number of elements in the set). So there has to be the same number of ordered pairs with C … Read more

¿A Dedekind-infinite set XX has a countable subset, and a proper subset equipotent to XX, but both with intersection empty?

In a more detailed way, Definition: A set X is said to be Dedekind-infinite if it has a proper subset A equipotent (similar/in bijection) to X. Let X be a Dedekind-infinite set. Does it exists A,B, subsets of X such that, A∼X, B∼N and A∩B=∅? If not, what conditions are needed to ensure this? In … Read more

(Proof) Order of Cardinals : $a\le c$, $b\le d$ $\rightarrow$ $a+b\le c+d$

claim $\forall \;set \; A,B,C,D$ Let $a =card(a), b=card(b), c=card(C), d=card(D)$ then $a\le c, b\le d$ $\Rightarrow$ $a+b\le c+d$ Proof $\exists \text{ injection } f: A \rightarrow C $ since $a\le c$ and similarly, $\exists \text{ injection } g: B \rightarrow D $ since $b\le d$ Then construct function $h : A\cup B \rightarrow C\cup … Read more

A function $f$ having , for all $k \in N$, the subsets of $A$ given by the solutions to $f(a)=k$ is finite, show that $A$ is countable or finite.

Could not solve this question, could anyone discuss it with me please? Answer Let $A_k = \{a\in A : f(a)=k\}$ so by assumption $A_k$ is finite. Now $A=\bigcup_{k=1}^\infty A_k$, it follows that $A$ is a countable union of finite sets and therefore is finite or countable. AttributionSource : Link , Question Author : Emptymind , … Read more