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.

learn more… | top users | synonyms (1)

0
votes
1answer
32 views

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 $|B|=\lambda$. ...
0
votes
3answers
67 views

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 ...
0
votes
1answer
11 views

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.
0
votes
1answer
37 views

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 ...
0
votes
2answers
21 views

Is this proof about equicardinality correct and/or rigorous? Can it be helped?

Here's the proof than a Cartesian product of two countable sets is countable(the proof is used, for example, in C.Pugh's "Real Mathematical Analysis" with one exception: they prove equicardinality of $...
0
votes
2answers
42 views

Proving cardinality of an uncountable sum

I'm trying to prove the following thing: For a family $\mathcal{A}$ of countable sets such that $|\bigcup\mathcal{A}|$ is uncountable and such that $\big|\{A\in\mathcal{A}: x\notin A\}\big| \leq\...
0
votes
1answer
21 views

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable.

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable. I don't really understand this statement. How can I see that the free product of countable $\...
0
votes
1answer
28 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. $X_{\alpha}...
0
votes
2answers
49 views

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 ...
0
votes
1answer
46 views

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 ...
0
votes
1answer
19 views

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 ...
0
votes
1answer
50 views

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 ...
0
votes
2answers
60 views

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: ...
0
votes
1answer
28 views

Cardinality of the set of functions which holds the Equality

Let $f$ be a function from $\{1,2,3, \dots ,10 \}$ to $\mathbb R$ such that $$\bigg( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\bigg)^2 = \bigg( \sum_{i=1}^{10} |f(i)|^2 \bigg) \bigg(\sum_{i=1}^{10} \frac{1}{...
0
votes
2answers
46 views

Prove that $\mathbb{|Q| = |Q\times Q|}$

I have this problem: Prove that $\mathbb{|Q| = |Q\times Q|}$ I know that $\mathbb Q$ is countably infinite. But then how can I prove that $\mathbb{|Q\times Q|}$ is countably infinite? Thanks ...
0
votes
1answer
54 views

Showing that $|\mathcal{P}(\mathbb{N})| = |\operatorname{Maps}(\mathbb{N},\mathbb{N})|$

Show that the cardinality of a power set of natural numbers is exactly equal to the map of a set of natural numbers to another, which is $|\mathcal{P}(\mathbb{N})| = |\operatorname{Maps}(\mathbb{N},\...
0
votes
2answers
43 views

Cardinal arithmetic confusion: What is $|\Bbb R|+|\Bbb N|$?

I do not understand how to calculate addition two cardinals. I know that the formula as follows: if $\alpha$ and $\beta$ are two cardinals, then $\alpha + \beta= |\{(a,0):a\in \alpha\}\cup\{(b,1):b\...
0
votes
1answer
62 views

A set of $\{ A, B , C , D , E\}$ with a cardinality of 3 [closed]

Struggling in my Discrete math class, and working on this problem I've read the notes but i am lost on a few things. On the first part, I am loss between to use the combination formula of $n!/(n-k)k!...
0
votes
1answer
35 views

Infinite Cardinal number power.

Let $a$ and $b$ are two infinite cardinal number than can i say that $a^{b}=2^{b}$? I am thinking so because of there this true for $\aleph_{0}$ and $c=2^{\aleph_{0}}$ as $\aleph_{0}^{c}=2^{c}$ and ...
0
votes
1answer
76 views

Cardinal number for a subset of $\mathbb{N}$

Following simple statement came to my mind when I was thinking about infinite sets. Statement: There is no set $X\subset\mathbb{N}$ that has cardinality strictly between any finite set $S\subset\...
0
votes
2answers
138 views

What is the cardinality of the power set $P(A \cup B)$

Let $A = \{1, 3, 5\}$ and $B = \{3, 4, 5\}$ be sets. What is the cardinality of the power set $P(A \cup B)$? If i'm not mistaking isn't it all the possible combination of these two: $\{\}, \{1,3\...
0
votes
2answers
84 views

The number of subsets of cardinality less than $\kappa$ of a cardinal $\kappa$ is $\kappa$

Let $L=\bigcup_{\alpha \in Ord} L_\alpha$ be Godel's constructible universe and thus $L \models GCH$. Let $\kappa$ be an infinite cardinal and $S:=\{A \subseteq \kappa : \#A < \kappa \}$. Is it ...
0
votes
1answer
44 views

Cardinality of Sets and injections

Let A,B,C,D sets. if |A| $\le$|B| and |B| < |C|, show that |A| < |C| Proof: Case1: suppose |A| < |B| then there exists injection f: A$\to$B and |B| < |C| then there exists injection ...
0
votes
3answers
94 views

Can we simplify analysis by getting rid of the uncountable reals? [duplicate]

Since the entire observable part of the universe can only be in a finite number of physically distinguishable states, it seems rather strange that an efficient formal description of the universe would ...
0
votes
2answers
41 views

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

As the title suggests, the question is : What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ? I'll tell you where im stuck, let's say f is ...
0
votes
1answer
78 views

Is the cofinality function monotonic?

Is the cofinality function $\operatorname{cf}$ monotonic? I.e., if $\lambda \le \kappa$ for cardinals $\lambda$ and $\kappa$, does it then follow that $\operatorname{cf}(\lambda) \le \operatorname{cf}...
0
votes
1answer
75 views

How could I prove that the cardinality of the union of two sets is equal to R? $|T U S| = |T| = |\mathbb{R}|$

I have to prove that $|T \cup S|$ where $T$ is infinite and $S$ is countable, equal to $|T|$, and this is also $|\mathbb{R}|$. How can I approach this? $|T \cup S| = |T| = |\mathbb{R}|$ I tried to ...
0
votes
1answer
21 views

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say $|A^\alpha|=\alpha$ ?

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say that the cardinality of $A^\alpha$ ($A \times A\times ...$ $\alpha$ times ) is $\alpha$ ? When $\alpha > |A|$ , ...
0
votes
2answers
95 views

Showing the set of functions $\{0, 1\} \to \mathbb{N}$ is countably infinite.

I'm doing a question it asked me to show that $\mathbb{N} \times \mathbb{N}$ was countably infinite but I am stuck on the following part of the question: deduce that the set of all functions $f : \...
0
votes
1answer
34 views

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t = c$ then $\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}} = c$

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t$ is equal to cardinality $c$, then $\:\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}}$ is also equal to cardinality $c$...
0
votes
2answers
72 views

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$?

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$ ?
0
votes
3answers
55 views

determining the cardinality [duplicate]

Let $S$ be the collection of closed intervals in the real line whose lengths are positive rational numbers. Determine the cardinality of $S$. Justify your answer As I understand, $S$ will be an ...
0
votes
2answers
27 views

How do I prove an equivalence of these two statements about Cantor's hypotheses?

How do I prove that the continuum hypotheses as stated by George Cantor (There are no sets with cardinality between the cardinality of the real and the cardinality of the rational numbers) is ...
0
votes
1answer
37 views

smallest cardinal greater than an infinite ordinal is a regular cardinal

let $\alpha$ be an infinite ordinal, and $\alpha^+$ be the smallest cardinal greater than $\alpha$. Show that $\kappa^+$ is a regular cardinal. This is for homework, but I'm not really sure where to ...
0
votes
1answer
40 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then $\aleph_{\alpha+1}^{\aleph_{...
0
votes
1answer
81 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that: $$\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $$ However I found this ...
0
votes
1answer
16 views

Building a function with codomain equal to a given set of reals.

I was discussing with friends the astounding fact that $\mathbb R$ and the set of real continuous functions were equipotent. I asked for a proof that $\mathbb R$ and $\mathbb R ^{\mathbb R}$ are not ...
0
votes
3answers
56 views

Understanding the proof of: If $|A| = \kappa$, then $|\mathcal{P}(A)|=2^{\kappa}$.

I don't quite understand the part where the author writes that $f$ is a one-to-one correspondence between the power set of $A$ and the set of all mappings from $A$ into $\{0, 1\}$. Isn't the function $...
0
votes
1answer
72 views

Cardinal Arithmetic, Regular Cardinals, and Exponentiation

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as well?...
0
votes
1answer
69 views

What is the cardinality of the set of all higher order functions mapping real functions to real functions?

What is the cardinality of the set of all higher order functions mapping real functions to real functions? To be specific, this set includes all higher order functions with the type signature: $(\...
0
votes
1answer
42 views

Find the number of vertices in the graph

Let $n\ge 1$ and $V_n = (\left\{ 1,2,...n \right\}\rightarrow\left\{ 0,1,2 \right\})$. Let us define $G_n = \left<V_n, E_n \right>$. $f,g$, are two vertices. They are connected iff: $$\left|\{ i ...
0
votes
1answer
44 views

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 ...
0
votes
2answers
291 views

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 $\#B = \#A$....
0
votes
2answers
49 views

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 ...
0
votes
1answer
66 views

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 ...
0
votes
1answer
116 views

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.
0
votes
2answers
84 views

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) ...
0
votes
2answers
56 views

$[\![n]\!]\times[\![m]\!]\sim[\![nm]\!]$ 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 $B$, ...
0
votes
1answer
31 views

Name for the sense of how many items are present

Sorry, this might be slightly off topic: there's a word for the ability to look at a small set of items are know how many are there without counting them, but I can't remember what it is and I can't ...
0
votes
2answers
58 views

$[(|X| \ge \aleph_0) \;\wedge\; (A \subset X) \;\wedge\; (|A| = |X\backslash A|)] \Rightarrow |A| = |X|$

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, $...