# Tagged Questions

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### If $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$

I've been thinking about the following claim: Let $A$ be a set and $|A|$ his cardinality. For every cardinal $\lambda$ with $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$. ...
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### Which sets have cardinal number $\aleph_{0}$ or $\mathfrak{c}$?

(a) $[1,3)$, $\mathfrak{c}$ (b) $Z$, $\aleph_{0}$ (c)$R \times R$, (d) $R \cap Z$, (e) $\{ 2^{-k} : k \in \mathbb{N} \}$ I understand that aleph null means that it is infinite and that c means ...
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### Cardinal Inequality without using Choice

Let $k$ be an infinite cardinal (i.e. not in bijection with any natural number), show that $\mathbb{N} \leq 2^{2^k}$ This is very easy with choice, without it I don't even know where to start.
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### If $\bigcup N_\alpha$ is stationary, then $\{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $\kappa$ and a disjoint family ...
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### Cardinal of an infinite set

In our course about combinatorics, our maths teacher recently introduced to us the notion of cardinality with the following definition: Let $E$ be a set. If there exists an integer $n$ and a ...
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### Need a formal proof?

If A and B are two equipotent sets (they have 1-1 correspondence). Prove that if A is denumerable then B is also denumerable. It is easy to understand by intuition. But I can't understand how to ...
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### Cardinality Proof Problem

Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A. I know that there is only an injection from A to B. I'm having trouble showing that ...
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### The cardinal number [closed]

Let $c$ be the cardinal number of $[0,1]$, i.e. $|[0,1]|=c$. Notice that $|A|\cdot|B| = |A\times B|$ and $|\mathbb{R}| = c$. Prove that $c\cdot c=c$. Don't use $ab=\max\{a,b\}$ where $a,b$ are ...
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### Show that the set is not countable

To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example: ...
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### A question about Choice Functions.

Assume the axioms of ZFC. Suppose that X is an infinite set of infinite (and pairwise disjoint) sets, none of which has a cardinal number greater than that of X. Is the cardinal number of the set of ...
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### Cardinality of two sets cross-multiplied

Let $A$ and $B$ be sets. Prove that $\#(A \times B) = \#(B \times A)$. What I have done: There exist an element $m$ in $A$ such that the element also exists in $B$. If $\#A = \#B$, then $\#B = \#A$....
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### The Cardinality of infinite series of natural numbers?

Given an infinite sequence $a_1,a_2,a_3,...$,and the map $F(a_1,a_2,a_3...) = {p_1}^{-a_1}{p_2}^{-a_2}{p_3}^{-a_3}...$ Where $p_i$ is the ith prime (chosen by the axiom of choice). Why isn't this ...
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### Proof Check: The cardinality of the set of all binary series with an infinite amount of 0's and 1's:

Label the set of all binary series with an infinite amount of 0's and 1's as $C$. It's easy to prove that the set (labeled $A$) of all binary series with a finite number of 1's is countable. I can ...
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### Subset of an infinite set with same cardinality

Let $A$ be an infinite set. Show that there is a subset $B\subseteq A$ such that $|B|=|A-B|=|A|$. I've tried using Zorn's lemma with $P(A)$ and $\subset$ and got nowhere. I would like a hint.
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### Elementary set theory - are these sets empty? [duplicate]

we are asked to answer if the following statements are true or false, and why: 1) The set ${\{\emptyset\}^{\mathbb N}}$ has exactly $1$ element. 2) The set ${{\emptyset}^{\mathbb N}}$ is empty. 3) ...
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### $[\![n]\!]\times[\![m]\!]\sim[\![nm]\!]$ where $[\![n]\!] = \{1,\ldots,n\}$

We have got: Let $n,m\in \Bbb N$ and denote $[\![n]\!]=\{1,\dots,n\}\subseteq \Bbb N$. Prove that: $$[\![n]\!]×[\![m]\!]\sim[\![nm]\!]$$ So conclude that, for the finite sets $A$ and $B$, ...
### $[(|X| \ge \aleph_0) \;\wedge\; (A \subset X) \;\wedge\; (|A| = |X\backslash A|)] \Rightarrow |A| = |X|$
Suppose that $X$ is an infinite set of cardinality $\alpha$. Also, suppose that, for some $A \subseteq X$, we have that $|A| = |X\backslash A|$. I want to show that $|A| = |X|$. When, for example, \$...