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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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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2answers
54 views

Prove the next cardinal property: $\kappa>1 \implies \lambda \leq \kappa^{\lambda}$ [closed]

Let $\kappa>1$ and $\lambda$ be cardinals. Prove that $\lambda \leq \kappa^{\lambda}$.
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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 ...
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2answers
282 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 = ...
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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 ...
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1answer
64 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 ...
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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.
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2answers
83 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) ...
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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 $B$, ...
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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 ...
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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, ...
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1answer
80 views

How to understand $f$ is one-to-one in Jech, 'Set theory' Lemma 3.10.?

I saw T. Jech, 'Set theory' Lemma 3.10: An infinite cardinal $\kappa$ is singular iff there exists a cardinal $\lambda<\kappa$ and a family $\{S_\xi :\xi<\lambda\}$ of subsets of $\kappa$ ...
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1answer
86 views

Cardinalties of Reals and Naturals

How can I show $n^{\aleph_0} = \mathfrak c$, if $n$ is finite and at least $2$? Also how can I show $\aleph_0 ^{\aleph_0} = \mathfrak c?$ I know there is a theorem thats says ...
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2answers
747 views

Does there exist a set of all cardinals? [duplicate]

Does there exist set that contains all the cardinal numbers?
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1answer
152 views

At least two circles meeting these cond. have nonempty intersection

Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets $\mathcal {S}$ is a family of circles in the plane such that for any $x \in \mathbb{R}$ there ...
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1answer
164 views

Mapping: each nonempty finite subset of $\Bbb R$ - sum of its elements

Could you give me a hint how to solve this problem? Let $ D:= \left\{E \subset \mathbb{R} \ | \ 0< \mathrm{card}(E)< + \infty \right\} $. $\phi\colon D\to\mathbb R$ defined by $\phi(E)\sum_{x ...
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1answer
129 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
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1answer
813 views

Is the set of integer coefficient polynomials countable? [duplicate]

Possible Duplicate: Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable? The set of integer coefficient polynomials are countable, when the cardinality of each set of ...
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1answer
94 views

If $a, p$ are cardinals satisfying $2 p = p$ and $a+p=2^p,$ then $a \ge 2^p.$

Theorem: If $\mathfrak a$ and $\mathfrak p$ are cardinals satisfying $2\mathfrak p = \mathfrak p$ and $\mathfrak a + \mathfrak p=2^\mathfrak p$, then $\mathfrak a \ge 2^\mathfrak p$. Here's a ...
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1answer
121 views

How to prove that $2^\omega=\mathfrak{c}$?

Let $\mathfrak{c}$ denote the continuum. My textbook says that $2^\omega=\mathfrak{c}$. How can one prove this equality? Thanks ahead:)
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1answer
78 views

how can we prove that $\cup C$ is countable? [duplicate]

Possible Duplicate: countably infinite union of countably infinite sets is countable proof that union of a sequence of countable sets is countable. I'm a newbie who try to understand Set ...
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2answers
304 views

Enumeration of real 'sequences', cardinality, Cantors diagonal argument.

Let us fix a well ordering of the real numbers then consider a 'list' of some subset of the real numbers (with at least two elements) - called A -, enumerated by the well ordering. Say our well ...
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1answer
18 views

Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where ...
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1answer
31 views

Connected Linear Graph not Path-Connected

Given a set of vertices $\{x_\alpha\}$ whose cardinality exceeds $\aleph_1$, (assume the axiom of choice) connect each vertex with its successor by an edge, forming a linear graph. Choose two vertices ...
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1answer
28 views

Maximal Sets and Bijections

I'm struggling with this question (The function $f(x) = x^2 -3$): Let $A = \{x \in R : x \geq 0\}$. Determine a maximum set $B$ such that $f : A \rightarrow B$ is a bijection. Let $g : B \rightarrow ...
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1answer
19 views

Constructing $2^\kappa$ vectors with a certain property

Take an infinite rectangular array with $\kappa$ columns and $2^\kappa$ rows where $\kappa$ is some infinite cardinal. Can you fill it up with at most $\kappa$ different elements in such a way that ...
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2answers
12 views

Finding the cardinality of a collection of lines?

I'm trying to find the cardinality of the set of lines such that the lines are not vertical, the slopes are a natural number, and the line has a natural number for a $y$-intercept. Because there are ...
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1answer
13 views

Showing the cardinality of a bounded shape in the xy plane?

I am trying to show that the cardinality of the space between $x^2+y^2<1$ and $x+y>1$ is the same as the cardinality of real numbers I haven't a clue where do begin with something like this. I ...
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1answer
37 views

Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$ for some $a,b,c,d\in\mathbb{R}$

Let $a,b,c,d\in\mathbb{R}$ such that $a<b$ and $c<d$ be given. Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$. My intuition is to construct a function ...
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1answer
44 views

What result is $\left | \bigcup_{i \in I}A_i \right | =\sum_{i \in I} |A_i|$?

I'm reading a text that uses the following equality for disjoint sets $(A_i)_{i \in I}$: $$\left | \bigcup_{i \in I}A_i \right |=\sum_{i \in I} |A_i|$$ This has to do with disjoint unions, but I'd ...
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2answers
42 views

Cardinal of a difference of power sets

How can the following be calculated? Given the sets $X = \{1, 2, \dots, 10\}$ and $Y = \{1, 2, \dots, 12\}$, compute $| \mathcal P (Y) \setminus \mathcal P (X) |$, where $\mathcal P (X) = \{ A \mid A ...
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1answer
42 views

Prove that $(0,1) =c\mathbb R$

How do I prove this knowing that $f(x) = \tan(x\pi/2)$ is a bijection between $(0,1)$ and $(0, \infty)$? We also have a bijection between $(-1,1)$ and $(0,1)$.
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1answer
35 views

Corollary 6.6 in Munkres' TOPOLOGY 2nd ed: Subsets of finite sets are finite? Cardinality of a proper subset is less than that of the set?

Here's Corollary 6.6 in Topology by James R. Munkres, 2nd edition: If $B$ is a subset of the finite set $A$, then $B$ is finite. If $B$ is a proper subset of $A$, then the cardinality of $B$ is less ...
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1answer
40 views

Cardinality of the set of functions from N to {0,1,2,3,…,9} is equal to card(R)

I know that the set of infinite sequences on {0,1,2,3,4,5,6,7,8,9} is uncountable, but how to show that it has a bijection to R?
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1answer
41 views

Find the cardinal of $X=\{A\subseteq \Bbb N : A \text{ is finite } \}$

Find the cardinal of $X=\{A\subseteq \Bbb N : A \text{ is finite } \}$. Attempt $$ X=\bigcup_{i\in\Bbb N}X_i$$ Where $X_i$ denotes the set of subsets of $\Bbb N$ with cardinal $i$. Let $A\in X_i, ...
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2answers
91 views

Trouble with definition of countable, denumerable

I found the following definition: Definition. A set is countable iff its cardinality is either finite or equal to $\aleph_0$. A set is denumerable iff its cardinality is exactly $\aleph_0$. A ...
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1answer
26 views

Finite Sets Proof on Domains [duplicate]

Just wanted some help with this little proof.: Let X and Y be Finite Sets. Prove that |X^Y| = |X|^|Y|
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1answer
82 views

Suppes' Axiom of Cardinal Numbers

In Suppes' book, Axiomatic Set Theory he introduces an axiom concerning cardinal numbers,before introducing them, namely that each set is associated with an object known as a cardinal number, and that ...
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1answer
25 views

bijective, one-to-one, and number of elements

How does one reconcile the following (seemingly) contradiction in using "number of elements" argument? In the "range" [0,1] in R there are more points than in N, to be shown as "take the inverse of ...
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2answers
53 views

Cardinality of the set of all complex sequences converging to zero.

I was asked to show that the set of all complex sequences converging to zero has the same cardinality as the set [0,1]. This is the only hole in a proof that I am working on. I need to show there ...
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1answer
48 views

Prove $A$ is either finite or countable.

Let $A$ be a set and let $f:A \to \Bbb{N}$ be an injection. Prove $A$ is either finite or countable. How can one prove such a thing? Since this is so obvious it gets me in a place where I don't know ...
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1answer
150 views

Cardinality of the set of all (real) discontinuous functions

This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6. (Show that) The cardinality of the set of all discontinuous functions is ...
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1answer
41 views

If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma=\beta^\gamma$.

Can someone suggest a rigorous proof of the following: If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma = ...
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1answer
30 views

Prove that if A~B then Sym(A)~Sym(B).

I tried to prove it with sets. Really, truly clumsy. I know |A|=|B|. Can I simply conclude that |A|!=|B|! => Sym(A)~Sym(B)?? (Sym(A) for a set A is the set of all bijections from A to A.)
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1answer
40 views

Transfinite fixed points of a function

Let the function $F\colon On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha): \alpha ...
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1answer
32 views

a question concerning multiplication of cardinal numbers

Consider $\{B_i\}$ where $i\in I$ and $I$ is countable infinite. $|B_i|=|B_j|=n$ for all $i,j$ and $n \ge |\mathbb{N}|$. I want to show that $| \large \cup_{i\in I}$$B_i|=n$ I am given that $a*a=a$ ...