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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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35 views

For cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$

I need to prove, without assuming the Axiom of Choice, that for cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$. I have already proved that for cardinals $c,c',d,d'$, if $c\neq0$,...
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0answers
18 views

Proving the sum of two finite cardinal numbers is finite

I am trying to prove the following: If $m,n\in \omega$, then $m\oplus n<\omega$ My proof is as follows Say wolog $m\in n$ Let $$f:(\{0\}×m)\cup (\{1\}×n)\rightarrow 2×\omega$$ $$f(t, p)=<...
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1answer
53 views

Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection [duplicate]

Question: Are there $2^{\aleph_{0} }$ sets of natural numbers such that each two have finite intersection. From what I've read about infinite families, I need to ignore those who have the properpty $...
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1answer
48 views

Indexing of uncountable sets and uncountable collections of sets, uncountable intersections containing a point

Definitions Let $\mathcal{A}$ be an uncountable collection of sets so that if $I_{\mathcal{A}}$ is the index set of elements of $\mathcal{A}$ then $|I_{\mathcal{A}}|\not=\aleph_{0}$ (I mean this to ...
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0answers
29 views

How to fill a rectangle with smaller specific rectangles that have cardinal information about their adjacent neighbours

Lets say its 6x6 grid that is represented by top left(0,0) and bottom right(1,1) in coordinate system. Next, I have set of objects with their cardinal directional information about each of their ...
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2answers
53 views

The cardinality of all equivalence relations over $\mathbb{N}$ [duplicate]

Let $R$ be the set the contains all equivalence relations over $\mathbb{N}$. Prove that $\left | R \right | = 2^{\aleph_0}$ This question is very counter - intuitive to me. I know that Each $R_i \...
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1answer
51 views

Finiteness of sets

For each of the following sets, determine if it is finite, countably infinite, or uncountable. You need not prove your answer is correct, but you should give a reason for your response. For some $n\...
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1answer
51 views

Countability of decimal representations of real numbers

Let $X=\{x\in \mathbb{R}\ | \ \hbox{the decimal representation of $x$ contains only 4s and 7s}\}$. Is $X$ countable or uncountable? Prove that your answer is correct. Should an argument like Cantor's ...
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0answers
39 views

Cardinality of partitions of a number

A partition of $n$ is a sequence of integers $(a_1, a_2, \dots, a_k)$ such that $a_i\geq 0$ for each $i$, and $\displaystyle\sum_{i=1}^k a_i = n$. The number $k$ is called the number of parts of the ...
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2answers
48 views

Cardinality of an arbitrary interval of real numbers

Let $a, b\in\mathbb{R}$ with $a<b$. Prove that $|\{x\in \mathbb{R}\ | \ a< x< b\}|=|\{x\in \mathbb{R}\ | \ 0<x<1\}|$. Would constructing a bijection be the most effective way to prove ...
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1answer
18 views

to show cardinality in separable Hilbert space setting

enter image description here the above exercise in Conway's Functional Analysis book to me, the setting is too rough so i have no idea to step forward could you help me to start this proof? or just ...
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0answers
41 views

Functions between ordinals.

I'm trying to compute the cardinality of a determined set. $$A=\{f\colon \omega_n \to \omega_m | |Supp(f)|=\aleph_k \quad k<n\}$$ As suggested by the exercise, I first tried few elementar cases: $$...
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1answer
38 views

Aronszajn tree for infinite singular cardinal [duplicate]

I've always seen Aronszajn trees being discussed on regular cardinals : Let $\kappa$ be a regular infinite cardinal. A $\kappa$-Aronszajn tree is a tree on $\kappa$ of height $\kappa$, whose ...
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1answer
22 views

Bijection between $ \bigcup_{i \in [0, 1]} X_i \ \ \ \text{and} \ \ \ [0, 1] \times [0, 1] $

So I have got the following sets $$ \bigcup_{i \in [0, 1]} X_i \ \ \ \text{and} \ \ \ [0, 1] \times [0, 1] $$ where each $X_i$ has cardinality $c$ of the continuum and each pair of $X_i$ where $i \in [...
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1answer
29 views

$X$ is defined as the collection of sets $X = \{X_i: i \in [0, 1] \}$ where each set has cardinality $c$

$X$ is defined as the collection of sets $X = \{X_i: i \in [0, 1] \}$ where each set has cardinality $c$ of the continuum and each pair is disjoint. How do I prove that for each $i \in [0, 1]$, the ...
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0answers
57 views

An uncountable set has uncountably many co-countable subsets containing an arbitrary point

This theorem seems "obvious" to me, but I want to check my logic since I am un-familiar with un-countably infinite sets, and I know these can give rise to non-intuitive results. Any comments welcome. ...
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2answers
22 views

Let $g: P \mapsto Q$ and $h: R \mapsto S$ be bijective.

Let $g: P \mapsto Q$ and $h: R \mapsto S$ be bijective functions. Give a bijection $X: R^P \mapsto S^Q$ where $M^N$ means the set of all functions from $N$ to $M$. I think I am missing something. I ...
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1answer
51 views

Formally representing “arbitrarily large” as a distinct entity from “countably infinite”

There is often an important difference to be made between "countably infinite" things and "arbitrarily large finite" things. There are so many examples of this that I need not list them all here. The ...
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2answers
40 views

Size of infinte sets cardinality

The question is as follows: Prove that if R is uncountable and T is a countable subset of R, then the cardinality of R\T is the same as the cardinality of R. What i have: I know that R is ...
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0answers
42 views

example of algebraic variety with infinitely many singularities

Let $X$ ba an algebraic variety and $\mathrm{Sing}(X)$ be the set of all singular points. For a set $A$ , $|A|$ denotes the cardinality of $A$ . I konw examples of algebraic variety with finite ...
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3answers
107 views

Cardinality of $\aleph _0$

I just started learning about cardinality in my math fundamentals class, and in particular I've just learned that $\aleph _0$ can be thought of as an equivalence class as follows: any set with ...
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1answer
30 views

Cardinality of permutations definition

I'm trying to understand the definition of cardinality of permutations through a basic example. If, for example, there is a set: $A = \{2, 3, 4, 2, 1\}$ What is the cardinality of its permutations? ...
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1answer
49 views

Subdirectly irreducible algebra of language F with cardinality $\geq 2^\kappa$ [closed]

If language $F$ has cardinality $\kappa$ ($\kappa$ is some infinite cardinal) and arity of every operation in F is 1, then doesn't exist subdirectly irreducible algebra of language F with cardinality $...
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1answer
22 views

Cardinality: Injection between subsets of Uncountable set

assuming, S is infinite uncountable, I am trying to come up with injective f: (S union N) -> S. Where N is naturals. So far I created S0 which consists of infinite sequence of elements of S, such that ...
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0answers
51 views

Cardinality of an uncountable set after union with another uncountable set, that has smaller cardinality.

When $A$ is uncountable and $B$ is a countable set, $|A\setminus B|=|A|$. How can I prove (or disprove) that $|A\setminus a|=|A|$ or $|A\cup a|=|A|$, where $A, a$ are uncountable sets and $|A|>|a|$ ...
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2answers
62 views

Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$

Using the fact that for any $A$, $A$ is countably infinite if there exists a bijection $f: A \rightarrow \mathbb{N}$, how do I prove the statement: $A$ is countably infinite if and only if $\exists$ ...
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2answers
66 views

Prove that $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$

I'm trying to prove the equality between the cardinalities: $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$ ($\mathbb{N}\times \mathbb{R}$ is cartesian product from $\mathbb{N}$ to $\mathbb{R}$). I ...
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0answers
33 views

linearly ordered family of sets cardinality greater than supremum of individual sets

Does there exist a set $X$ linearly ordered by $\subset$ where $|\bigcup X|>\sup\{|x|:x\in X\}$? I'm having trouble thinking around constructing one. I'm fairly certain that finite sets won't work,...
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1answer
77 views

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? [duplicate]

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection ...
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1answer
118 views

Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable

Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$. I came ...
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1answer
39 views

Show $\left| \{H,T\ \}^{\oplus \mathbb{N} }\right| = \left| \mathbb{R}^{\mathbb{N} }\right|$

Let $\mathbb{R}^{\mathbb{N}} = \{ f: \mathbb{N} \to \mathbb{R}\}$ with the product $\sigma$-algebra $\mathcal{B}^{\otimes \mathbb{N}} = \bigotimes\limits_{n \in \mathbb{N}} \mathcal{B}$, where $\...
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1answer
40 views

$\kappa\cdot\sum_{i\in I}\lambda_i=_c\sum_{i\in I} \kappa\cdot \lambda_i$

Moschovakis Exercise x4.20 Prove that for all indexed families of cardinals, $$\kappa\cdot\sum_{i\in I}\lambda_i=_c\sum_{i\in I} \kappa\cdot \lambda_i$$ We have $$\kappa\cdot\sum_{i\in I}\lambda_i=...
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0answers
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$\lambda\le_c\mu\implies \kappa^\lambda\le_c\kappa^\mu$

Moschovakis, (part of) Exercise x4.16: Prove that for all cardinal numbers $\kappa,\lambda,\mu$ $$\lambda\le_c\mu\implies \kappa^\lambda\le_c\kappa^\mu$$ provided $\kappa\ne 0$. $\le_c$ means "...
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1answer
37 views

order type of club - cofinality

With $ot(C)$ being the order-type of the club $C$, and $cof(\alpha)$ the cofinality of the ordinal $\alpha$ Show that for every limit ordinal $\alpha$ there is a club $C\subseteq \alpha$ such that $...
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2answers
64 views

How can I prove that if $\alpha$ is an ordinal, then there is an initial ordinal $\kappa$ such that $|\alpha|=|\kappa|$?

I'm having trouble understanding initial ordinals. In particular, I can't prove a seemingly trivial theorem about them. Def: An ordinal $\kappa$ is an initial ordinal iff $\forall \delta < \kappa \...
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0answers
60 views

Proving $|A\cup B|\le_c |A|+|B|$

I'm using the notation from this question. For all sets $A,B$, $|A\cup B|\le_c |A|+|B|$, and if $A\cap B=\emptyset$ then $|A\cup B|=_c |A|+|B|$. My attempt to prove it: By definition, $$|A|+|B|=||...
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1answer
74 views

Problem of cardinal assignment

A weak cardinal assignment is any definite operation on sets $A\mapsto |A|$ which satisfies (C1) and (C3), and it is a strong cardinal assignment if it also satisfies (C2). The cardinal numbers (...
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0answers
18 views

Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
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1answer
19 views

S is a possibly infinite set. Prove |S| < |P(S)|

Suppose S is a set. Do not assume that S is finite. How can one prove that |S|<|P(S)|? (P(S) is the power set of S). Would I say something how the power set is the subset of S so that it contains ...
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1answer
41 views

Diagonal argument applied to computable numbers

Upon applying the Cantor diagonal argument to the enumerated list of all computable numbers, we produce a number not in it, but seems to be computable too, and that seems paradoxical. For clarity, ...
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0answers
39 views

for any infinite set A, |A| >= |N| [duplicate]

I try to prove that the power of any infinite set must be equal/bigger then the power of the natural א0. I tried to show that for any infinite set there exists a subset of it, that its power is the ...
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0answers
131 views

Two well orderings of an infinite cardinal agree on a large set

I've seen this question but I'm having trouble following the proof given. This is an exercise from Kunen: If $\kappa$ is an infinite cardinal and $\triangleleft$ is a well ordering on $\kappa$, then ...
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0answers
32 views

$\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
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1answer
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$X \subset \mathbb{R}_{>0}$ exists $C$ for every $\{x_1, …, x_n\} \subset X$ $\sum^{n}_{i=1} x_i<C$ Then $X$ is countable

Let $X \subset \mathbb{R}_{>0}$ such that exists $C$ that, for every $\{x_1, ..., x_n\} \subset X$ $$\sum^{n}_{i=1} x_i<C.$$ Prove that $X$ is countable My intuition goes that if $X$ is not ...
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4answers
67 views

Cardinality of $A$ and the Natural Numbers [closed]

The question asks to show that the set $A$ and the set of natural numbers have the same cardinality. Where $$A=(1,2,3)\times\mathbb{N}$$ So I know I have to prove a bijection, but I am having ...
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2answers
67 views

cardinality proof: prove that for any set $A, \ 2^{|A|} \neq |\mathbb{N}|$ [duplicate]

Prove: for any set $A, \ 2^{|A|} \neq \aleph _{0}$ as $\aleph_{0} = |\mathbb{N}|$. my attempt: Suppose by contradiction that there exist a set $A$ such that $2^{|A|} = \aleph_{0}$, which Implies ...
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1answer
54 views

Decomposition of an $\aleph_1$ set into $\aleph_1$ sets $\aleph_1$ [closed]

Can we justify the claim Any set of cardinality $\aleph_1$ can be expressed as the disjoint union of $\aleph_1$ sets of cardinality $\aleph_1$ in a simple yet reasonably-correct way?
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1answer
31 views

An ordinal equipotent to a natural number is equal to that number

Let $\alpha, \beta$ be ordinals and $n$ be a natural number. How can one prove the following $$ \forall \alpha (\alpha \approx n \implies \alpha = n)$$ using the fact $$ \vert \alpha \vert \le \beta \...
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1answer
39 views

what is the cardinality (or algebraic dimension) of a hilbert space over a cardinal $\kappa$?

Something has been bugging me lately; maybe you guys can help satisfy my curiosity. Let $\kappa$ denote any nonzero cardinal. For convenience, we identify $\kappa$ with its smallest ordinal ...
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1answer
199 views

Why is the definition of cardinal number as the set of all sets equivalent to a given set “problematical”?

In Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one-to-...