Questions tagged [cardinals]

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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1answer
19 views

If $B \subseteq A$ and $f:A \xrightarrow{1-1} B$, prove that $\bigcap_{n \lt \omega}f^n[A]=\bigcap_{n\lt \omega}f^n[B]$

If after confirming the validity of my proof, someone could comment on the possibility of creating a direct proof...that would be awesome. At any rate: Prove that $\bigcap_{n \lt \omega}f^n[A]=\...
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1answer
29 views

Proof that vector spaces $k^M$ and $k^{(M)}$ are not isomorphic

Assume $k$ is a division ring and $M$ an infinite set. There are two vector spaces over $k$ to consider: $k^M$ and $k^{(M)}$ (the latter is the subset of $k^M$ containing all infinite sequence with ...
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0answers
48 views

Show that f: ℤ => ℕ is surjective [closed]

I need to show that the function $f:\mathbb{Z}\to\mathbb{N}$ given by: $$f : k\mapsto \vert k\vert = \begin{cases} k &\mathrm{if}~k\geq 0 \\ -k &\mathrm{if}~ k<0\end{cases}$$ is a ...
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1answer
42 views

Cardinals proof |A| ≤ |B| iff |B| ≥ |A| [duplicate]

So i have a little bit problems proofing this. |A| ≤ |B| iff |B| ≥ |A| i know, that if ...
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1answer
27 views

Is the cardinality of the topology of a separable infinite metric space uncountable

Given a metric space $X$ which is separable and infinite, can we say that $\tau$ (its topology) has cardinality greater than that of the natural numbers? And if so how to go about it.
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1answer
55 views

Is there a one-to-one corrospondence between integer functions and the reals? [closed]

Assuming $\Bbb{Z}^\Bbb{Z}$ to mean the set of all functions from $\Bbb{Z}$ to $\Bbb{Z}$, are $\Bbb{Z}^\Bbb{Z}\cong\Bbb{R}$? Another way to put it, assuming a set of functions $Z:=\{f:\Bbb{Z}\...
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1answer
34 views

Prove that for $S$ a maximal mean less subset, $|S|=\aleph$.

We say that $S \subset \mathbb{R}$ is a mean less subset if for all $a \in S, b \in S$ we get that $(a+b)/2 \notin S$. Prove that for $S$ a maximal mean less subset, $|S|=\aleph$. Clearly we know ...
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1answer
27 views

Stuck with a proof regarding cardinality

Problem: For any set $A$, finite or infinite, let $B^{A}$ be the set of all functions mapping $A$ into the set $B=\{0,1\}$. Show that the cardinality of $B^{A}$ is the same as the cardinality of the ...
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1answer
23 views

Let $f : N → Y$ be a map having right inverse $g$. Prove that $Y$ is at most countable? [duplicate]

I know this implies $F$ is surjective, but not sure if this helps.
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0answers
18 views

How can we show that the set of all polynomials with integer coefficients denumerable?

Is the set of all polynomials with integer coefficients, denumerable? I know that this set is countable but how can I show that it is denumerable as well.
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1answer
101 views

Proof that $\mathbb{R}$ is not countable

I know that this proof may sound ridiculous, but I'm really curious to find out if it's logically correct(and whether there are some circularities). Since $|[0,1]|=|\mathbb{R}|$, we have to simply ...
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1answer
34 views

Exponentiation of infinite cardinals: rules involving $\max\{\kappa_1, \kappa_2\}$?

Th question is "simple". Let $\kappa_1, \kappa_2$ be two infinite cardinals. We have the simple rules for addition and multiplication: $$ \kappa_1+ \kappa_2=\kappa_1 \kappa_2=\max\{\kappa_1, ...
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1answer
48 views

Finite sets and cardinality.

Let $$ I_N = \{n \in \mathbb N \mid 1 \leq n \leq N \}=\{1,2,3, \ldots, N \}.$$I want to show that $I_n \times I_m \sim I_{nm}$. For that, I defined $ \psi: I_n \times I_m \to I_ {nm} $ by $$ \psi (...
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1answer
38 views

Intuition of countable sets using discrete sets

I'm trying to think of a way to intuitively explain myself what a countable set means. So far as I understand it, in very simple words, a set $S$ is countable iff you can "name" or "...
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2answers
91 views

Rank $ (a) {\aleph_4}^{\aleph_4}, (b) {\aleph_0}^{\aleph_4}, (c) {\aleph_4}^{\aleph_0}, (d) {\aleph_4}, (e) {\aleph_5}, (f) {\aleph_5}^{\aleph_0} $

If we work in a model of set theory where $2^{\aleph_0} = \aleph_5$, then Rank in ascending order the following cardinals: $$ (a) {\aleph_4}^{\aleph_4}, (b) {\aleph_0}^{\aleph_4}, (c) {\aleph_4}^{\...
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2answers
93 views

Let $T =\{X\in P(\mathbb{Q}) | X\cup \mathbb{N} = \mathbb{Q} \}$ the cardinality of $T$ is $\aleph$?

Let $T =\{X\in P(\mathbb{Q}) | X\cup \mathbb{N} = \mathbb{Q} \}$ the cardinality of $T$ is $\aleph$ ? prove attempt we know that $X\in P(\mathbb{Q})$ so, $X\cup \mathbb{N} = \mathbb{Q}$ create an ...
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1answer
38 views

A question about what does $|A|^{<\omega}$ mean for a set

I was reading this answer https://math.stackexchange.com/a/346249/629594 and I am not really sure what $|A|^{<\omega}$ means (I am not so familiar with things that concern ordinals and cardinals). ...
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0answers
29 views

What is the cardinality of the set of all real numbers in $(0,1)$ that have more than one representation?

for example $\frac 1 {100}=0.01000000...$ and $\frac 1 {100}=0.009999999...$ Intuitively, I think it's $\aleph_0$, and I tried to consider a function (let $A$ be this set), $f:A\rightarrow \mathbb{N}...
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2answers
51 views

Cardinality of an infinite tree paths

Suppose a tree with nodes located at levels $1,2,3...$. At each level the nodes branch into several nodes or do not branch. Does the cardinality of the set of all infinite paths in this tree depend on ...
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1answer
38 views

Counting question and cardinality

A football league contains $6$ teams. During the season each team plays two matches against each other team. The result of each match is a draw or a win for one or other team. How many matches are ...
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1answer
47 views

Bijection between $ (0,1) ^ n $ and $ (0,1) $.

Let $ z \in (0,1) ^ n $. We can express this "$ z $" as a $ n $ -tuple of real numbers $ z = (z_1, z_2, \ldots ,z_n) $, where each $ z_i $ will have its infinite decimal expansion. We can ...
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2answers
57 views

countable-to-one set cardinality

Let $B$ and $C$ be nonempty sets. Say a map $ \phi : B\to C$ is countable-to-one if for every $c \in C, \phi^{-1}(\{c\})$ is countable. Show that if such a map exists then $|B| \leq \aleph_0|C|.$ ...
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1answer
61 views

dimension of infinite dimensional vector space

Let $V$ be a vector space over a field $\mathbb{K}$ (either $\mathbb{C}$ or $\mathbb{R}$) that has an infinite linearly independent subset. Prove that if $B$ and $B'$ are two bases for $V,$ then $B$ ...
3
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1answer
124 views

Proof that non-well-orderable sets are not equinumerous to their own cardinals (using Scott's trick)

Azriel Levy's Basic Set Theory (the 2002 Dover edition) contains the following problem (Chapter 3, Exercise 2.24): Prove that $|x|$ is an ordinal iff $|x| \approx x$, (i.e., iff $|\,|x|\,| = |x|$). ...
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0answers
32 views

finding cardinality of set of all functions from Q to R

suppose $f:Q→R$, how we can find cardinality of set of all possible functions $f$,I know it's probably the same as cardinality of real numbers but I can't think of any bijection for that or any other ...
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1answer
36 views

finite subsets proof

If $A$ is an infinite set then $\mathcal{F}(A),$ the set of finite subsets of $A$ has the same cardinality as $A$. Is the following proof of the above fact correct? Note that $|A_k|\leq |A^k|$ for ...
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1answer
39 views

equivalent properties of cardinals

Let $A\neq \emptyset$ be a set. Show that the following are equivalent: $|\mathbb{N}| = \aleph_0\leq |A|$ $|A| = n|A|$ for any $n\geq 2\in \mathbb{N}$ $|A| = \aleph_0 |A|$ I want to show $(1)\...
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2answers
28 views

Cardinal Exponentiation Lemma

The following lemma is stated in Kunen's Set Theory: $\kappa^{\lambda}= 2^{\lambda} = |\mathcal{P}(\lambda)|$ whenever $\kappa,\lambda$ are cardinals with $\lambda$ infinite and $2 \le \kappa \le 2^{\...
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0answers
26 views

cardinal addition theorems

A cardinal number $\alpha$ is a number such that there exists a set $A$ so that $|A| = \alpha.$ The sum of two cardinal numbers $\alpha$ and $\beta$ is given by $\alpha + \beta := |A\cup B|,$ where $A$...
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1answer
42 views

Cardinality of Real numbers in arbitrary small interval. [duplicate]

I have been learning about the cardinality of Real numbers and it got me thinking. Since the cardinality of real numbers is uncountable and is the same as the cardinality of real numbers in the ...
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0answers
19 views

Can every uncountable set be expressed as a disjoint union of uncountable sets? [duplicate]

I want to show that the co-countable countable $\sigma$-algebra on $X$ is distinct from $P(X)$ whenever $X$ is uncountable. Can we express every uncountable set as the disjoint union of uncountable ...
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1answer
50 views

Choose whether or not the given sets are equinumerous to $\mathbb{N}^\mathbb{N}$.

Which of the following sets are equinumerous with $\mathbb{N}^\mathbb{N}$? (i) $\mathbb{N} $ (ii) $\mathbb{R}$ (iii) $2^\mathbb{R}$ (iv) $\mathbb{N} \times \mathbb{R}$ (v) $\mathbb{R}^\mathbb{R}$ (vi) ...
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0answers
47 views

Cardinality of $ \mathbb R \times \mathbb R $ and $ \mathbb R $.

My attempt is based on giving an explicit function between both sets but through compositions, which is up to what I have learned so far. First, let us consider the map $ f: (0,1) \times (0,1) \to (0,...
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0answers
14 views

Is a basis for the vector space of all series in $\mathbb{R}$ constructible [duplicate]

Given the $\mathbb{R}$-vectorspace $V =\mathbb{R}^\mathbb{N}$ of real valued series I was wondering if we can construct a Hamel-basis of $V$. First of all I think that the dimension of $V$ is $ \mid\...
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1answer
94 views

Proving $|P(\mathbb{N})| $ is equinumerous to these 2 sets.

$ \left\{A\in \:P\left(\mathbb{N}\right):\:\:\left|A\right|=\left|\mathbb{N}\right|\right\} $ All equivalence,homogenous relations over $ \mathbb{N} $ For the first set I managed to prove 1 sided ...
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1answer
67 views

Cardinality of the set of points on a circle whose coordinates are constructable is $\aleph_0$

Let $C=${$(x,y)\in\mathbb R^{2}|(x-a)^{2}+(y-b)^{2}=r^{2}$} be a circle of radius $r>1$ and centre $(a,b)$ where $a,b,r$ are all constructible. We want to show that the cardinality of the set of ...
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1answer
26 views

Counterexample to slight alteration of Theorem

One can quite easily prove that if $(A,<)$ is linearly ordered and for all $a\in A$, $$|\{b\in A\mid b\leq a\}|<\aleph_\alpha,\quad (*)$$ then $|A|\leq \aleph_\alpha$. The strict inequality in $(...
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1answer
56 views

Can we have a set that is hereditarily equinumerous to itself, of any size?

Is it consistent to add to ZFC-Reg. the existence of a nonempty set $\mathcal H_\mathcal H$ that is hereditarily equinumerous to itself? If that is consistent, then is it consistent that we can have $\...
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1answer
36 views

How to define a function to show that infinite sets are countable

To prove cardinality for infinite sets, I know that I have to show that there is a bijection. However, I'm having some trouble defining a function for my sets. For example: Let O be the set of all odd ...
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1answer
41 views

How to construct an increasing sequence in an ordinal $\alpha$ from a bijection $f: |\alpha| \rightarrow \alpha$

I have a question related to cofinality: Let $\alpha$ is a limit ordinal and is not a cardinal. How can I construct an increasing sequence $\alpha_{\xi}$ in $\alpha$ with the length $|\alpha|$ from a ...
1
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1answer
52 views

Is cardinal exponentiation strictly monotone in the exponent? [duplicate]

Let $\kappa$ be a cardinal, does $2^\kappa<2^{\kappa^+}$ always hold? It clearly holds if one assumes generalized continuum hypothesis, but does it also hold if one assumes otherwise?
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3answers
27 views

For sets A and B, where A is countable and B is uncountable, what would $A \setminus B$ be?

Since A is countable and B is uncountable then this would be undefined as B is larger than A right? Or am I missing something here?
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0answers
47 views

Bijection proof. Exercise

How can I prove that the cardinality of $ (0,1) ^ n $ is equal to that of $ (0,1) $, giving an explicit bijective function. I edit my question. What I am trying to achieve is to construct a bijective ...
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0answers
22 views

Injections and Cardinalities

A set $X$ is said to have equal cardinality with another set $Y$ iff there exists a bijection between $X$ and $Y$. Furthermore, a set $X$ is said to have cardinality $n$ iff there exists a bijection ...
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1answer
28 views

Cardinality of order preserving functions from totally ordered set with dense subset

I know that in the category of continuous functions, if $X$ and $Y$ are Hausdorff topological spaces and $D\subseteq X$ is a dense subset of $X$, then the cardinality of the continuous functions from $...
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1answer
45 views

Cardinal Arithmetic Property Proof

The following proposition is "obvious" but I am having trouble obtaining a rigorous proof of it: If the set $X$ is finite and $Y$ is a subset of $X$, then $Y$ is finite and #$(Y)\leq$#$(X)$. ...
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1answer
31 views

Cardinality of the set of functions that maps from $\mathbb{N}$ to $\{1,2,3\}$.

I know that from the other direction, the cardinality of the set of function that maps from $\{1,2,3\}$ to $\mathbb{N}$ is $\mathbb{N}\times\mathbb{N}\times\mathbb{N}=|\mathbb{N}|$, however, I don't ...
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2answers
23 views

finite sets and countable sets.

Let $ A \neq \emptyset $. If $ A $ is finite and $ B $ is countable, then $ A \times B $ is countable. My attempt is as follows, since $ A $ is finite and $ B $ is countable then $$ A = \{a_1, a_2, \...
0
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1answer
34 views

Ordinals and Cardinals larger than the fixed points of $α↦ω_α$ and $α↦\aleph_α$?

Surely there is no limit to how high we can go, so how do we talk about ordinals and cardinals higher than the fixed points of the functions $α↦ω_α$ and $α↦\aleph_α$? Is the power set of the $\aleph$-...
1
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0answers
28 views

Proving cardinality given surjection from A to B and B to A [duplicate]

Suppose f:A→B and g:B→A are both surjective, does this imply that there is a bijection between A and B. I was told that the statement above is true only with Axiom of Choice. Can someone provide an ...

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