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
66 views

Finding bijection from (0,1) → N

How exactly do I go about finding a bijection between (0,1) → N \ {0} so $(0,1) → (1, \infty)$. I figured I could look at this as finding a function from $(0,1) → (0, \infty)$ and just adding 1. ...
1
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1answer
54 views

Cantor's bijection between the sets [0,5] and [12,60]

How can I mathematically prove that the sets $[0,5]$ and $[12,60]$ have the same cardinality using Cantor's bijection? While simply drawing a linear function seems tempting, my teacher wants us to do ...
0
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1answer
38 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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3answers
43 views

I dont understand this sets question [closed]

Let $A = \{\{0\}\}$ and $B = \{0\}$. Which of the following statements are true and which are false? Justify each of your answers. $|A| = |B|$ (10 marks) $A \cap B = \emptyset $ (10 marks) $A \cap ...
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1answer
82 views

Show that the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$ [duplicate]

Show that $| \mathbb N | = | \mathbb N \times \mathbb N |$, i.e., the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$. How do I show it using the ...
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0answers
17 views

Investigating the cardinality of the set of all subsequences of any arbitrary sequence?

This is a non-precisely formulated question recently come to mind: How to investigate the cardinality of the set of all subsequences of an arbitrarily given sequence? Or can we possibly determine the ...
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0answers
26 views

double horizontal bars notation for sets

I am unsure of the meaning of a specific notation. We have a two-dimansional matrix $I_B$ (representing an image after applying Gaussian blurring to it, but that's just background information). Let ...
1
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1answer
13 views

cardinality of a countable sequence of powersets

Given a set $S_1$ of cardinality $\kappa$, we can construct the sequence $\langle S_1, S_2, S_3 ... \rangle$, where $S_i = \wp(S_{i-1})$, for all $i > 1$. If $S$ is finite, so that $\kappa < ...
4
votes
2answers
100 views

On singular products of cardinal numbers

I want to know whether it is possible to show in $ZFC$ that there exist a limit ordinal $\lambda$, a strictly increasing sequence of cardinal numbers $\langle \mu_\alpha : \alpha \in \lambda\rangle$ ...
0
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2answers
34 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 ...
4
votes
2answers
220 views

Is there more than one instance of the Empty Set?

It seems any additional instance would be equivalent in every respect to the first, hence indistinguishable, and arguably identical. I.e., there is only one Empty Set. Correct?
2
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2answers
30 views

If one of two sets has larger cardinality, there is a map onto the other set

Let A and B be sets with the cardinality of A less than or equal to B. Show there exists an onto map from B to A. I am struggling with this proof. I don't know how to show this. Any help would be ...
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2answers
28 views

The cardinality of the set

Let $\mathbb{G} =\{ a^b + \sqrt{c}: a,b,c\in \mathbb Q \}$ I guess the set $\mathbb{G}$ is countable set, but I can't show it properly. How to start the proof?
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2answers
61 views

König's theorem (set theory) implication

How does König's theorem imply $\quad\aleph_{\omega} \neq \beth_1$?
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7answers
404 views

Sets with same Cardinality, but no Explicit Bijection?

Are there any good examples of sets where we know that they have the same cardinality, but have not found any explicit bijection between them?
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3answers
200 views

The set of all functions from integers to a finite set is uncountable

Show that the set of functions from positive integers to the set $\{0,1,2,3,4,5,6,7,8,9\}$ is uncountable. I suspect I should use the diagonalisation argument but I'm not sure how to approach it. ...
2
votes
2answers
42 views

Proper Subsets and One-to-One

I want to make sure I am understanding this correctly. If we have a function $f$ which maps $A$ to $B$, and $A$ is a subset or proper subset of $B$ then $f$ is one to one from $A$ to $B$? Is that ...
5
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1answer
77 views

Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$?

Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$ ? I think I should use the Schroder-Bernstein theorem but I can't find the necessary injections/ prove that there aren't ...
0
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0answers
31 views

What is the cardinality of the set of all bijections from a countable set to another countable set? [duplicate]

What is the cardinality of the set of all bijections from a countable set to another countable set?
2
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1answer
89 views

Does there exist a bijection from $[0,1]$ to $\mathbb R$? [closed]

We can find a bijection from $(0,1)$ to $\mathbb R$. For example, we can use $f(x)=\frac{2x-1}{1+|2x-1|}$ composed of parts of two hyperbolas, see the graph here. Or we could appropriately scale the ...
0
votes
3answers
60 views

An uncountable subset of $\mathbb{R}$ contains a convergent sequence.

I was wondering whether it was true that an uncountable subset of $\mathbb{R}$ contains a convergent sequence. I was thinking about a proof by contradiction but did not manage to complete it. I ...
0
votes
1answer
59 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 ...
6
votes
3answers
484 views

Find a bijective mapping that shows that [0,1] and [0,1) have the same cardinality [duplicate]

I need to show that the two sets $[0,1]$ and $[0,1)$ have the same cardinality. I know that in order to show this I must show that there exists $f$ such that $f:[0,1]\to[0,1),$ but I am not sure how ...
2
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1answer
80 views

Follow-up question on Monotonic “Subfunction”

Let $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ be an arbitrary function. Must there exist $E\subseteq\mathbb{R}$ of cardinality $\aleph_1$, such that $f$ restricted to $E$ is monotonic? Assuming CH, ...
2
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0answers
74 views

Concerning the ring of all real valued functions of bounded variation on $[a,b]$

Let $B[a,b]$ the ring of all real valued functions of bounded variation on $[a,b]$ . What is the cardinality of $B[a,b]$ ? How does the maximal ideals of $B[a,b]$ look like ? How does the prime ideals ...
0
votes
1answer
24 views

Cardinality of the set difference.

If $B\subset A$, is it true that $Card(A)\backslash Card(B)$ is idempotent to $A\backslash B$ ? It seems to be true though I do not know how to prove it.
0
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1answer
48 views

Proving Cardinality of Sets

Prove that the sets E = {x ∈ ℕ : x = 2k, k ∈ ℕ } and ℕ have the same cardinality. A clear definition of cardinality was not given in this situation, so I understand cardinality to be (more or less) ...
0
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1answer
38 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, ...
1
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1answer
44 views

Question about cardinal arithmetic

Let $|I|\leq \aleph _\alpha$, and let $ \left\{ \beta _i \right\} _{i\in I}$ be cardinals with $\beta _i \leq \aleph _{\alpha}$. Is it true that $\aleph_{\alpha +1} > \sum _{i\in I}\beta _i $? ...
0
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0answers
37 views

Is one-to-one correspondence required for determining cardinality?

If I wanted to show that two sets had the same cardinality, would I have to show that they are a function in which the two sets are in one-to-one correspondence? For example, if I wanted to start a ...
0
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1answer
28 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 ...
1
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1answer
57 views

Comparability of vector spaces over the same field

Let $V$ be the collection of linear isomorphism classes of vector spaces over field $F$. Is this class $V$ totally ordered by inclusion? Argument (possibly erroneously): (The following assumes that ...
3
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2answers
45 views

Prove that if $X$ and $Y$ are sets where $\,\left|X\right|=\left|Y\right|,\,$ then $\,\left|P\left(X\right)\right|=\left| P\left(Y\right)\right|$.

We are basically being asked to prove that if the cardinality of set $\,X\,$ and set $\,Y\,$ is the same, then how can we prove that the cardinality of their power sets is also the same. I have ...
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1answer
37 views

Basic Facts About the Cardinality of the Power Set

If $X$ and $Y$ are nonempty sets, we define $Y^X = \{f\,|\,f\colon X\to Y\}$ and $\mathrm{card}(Y^X)=\mathrm{card} (Y)^{\mathrm{card}(X)}$. Show that the above definition is independent on the sets ...
3
votes
2answers
59 views

Give a bijection $f: (c,d) \to \Bbb R$ (f no trigonometric) to prove every open interval has the same cardinality of R [duplicate]

I want to prove that every open interval has the same cardinality of R. I've proved that $|(a,b)|=|(c,d)|$ so I may find a bijection $f: (a,b) \to (c,d)$. I need a bijection $f: (c,d) \to \Bbb R$ (a ...
3
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3answers
151 views

Prove: Any open interval has the same cardinality of $\Bbb R$ (without using trigonometric functions)

I want to prove that every open interval has the same cardinality of $\Bbb R$. The question is: Is it enough to prove that any open interval is uncountable? If I prove it, can I say that this ...
7
votes
1answer
202 views

Monotonic “Subfunction”

Let $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ be an arbitrary function. Must there exist $E\subseteq\mathbb{R}$ of size continuum, such that $f$ restricted to $E$ is monotonic? I guess this ...
0
votes
2answers
73 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 ...
3
votes
1answer
49 views

if $\alpha$ is an ordinal is it true that ${\aleph _{\alpha +1}}^{\aleph _{\alpha}}=\aleph _{\alpha +1}$?

If we denote the following cardinals: $\beta _0=\aleph _0$, $\beta _k=2^{\beta _{k-1}}$ then I know that ${\beta _{k+1}}^{\beta _k}=\beta _{k+1}$ but, is it true that for some ordinal $\alpha$, ...
0
votes
1answer
80 views

Proving $2^{\aleph_0} = {\aleph_0}!$ [duplicate]

How can I show the existence of a injection $\phi:\{x|x \subset \omega \} \rightarrow \{f|f:\omega \rightarrow \omega$ is bijective$\} $?
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1answer
110 views

Does there exist a bijection between sets $A\setminus B$ and $A$?

Let $A$ and $B$ non-empty sets, A is infinite and B is countably infinite($\sim \mathbb{N}$). Prove that if A is not countably infinite and $B\subseteq A$, then exists a bijection between $A\setminus ...
2
votes
3answers
117 views

Proving Power Set of $\mathbb N$ is Uncountable

I'm getting hung up on a proof that I remember being fairly easy... Showing that the power set of $\mathbb N$ is uncountable. Supposing it's countable, say $A=\{A_1,...\}$, we choose a set $B$ ...
3
votes
1answer
77 views

Explicit bijection between $[0,1)$ and $(0,1)$ [duplicate]

Proving that $[0,1)$ and $(0,1)$ have the same cardinality (without assuming any previous knowledge) can be done easily using Cantor-Bernstein theorem. However I'm wondering if someone can build an ...
3
votes
1answer
40 views

Existence of a partition of a set $X$ which contains at least 2 subsets of $X$ with the same cardinality

If $X$ is a set s.t $|X|\geq \aleph _0$ then I want to prove that there exist $A,B\subset X$ s.t $A\cap B=\emptyset$, $A\cup B=X$ and $|A|=|B|=|X|$. How can I construct such subsets?
6
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3answers
195 views

Prove: $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable

Prove: The set $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable I have this idea but I'm not sure if it's correct: We know $\Bbb Q$ is countable so we can list $\Bbb Q$ as $\Bbb Q= \{ ...
2
votes
1answer
68 views

Total ordering on $\mathcal P(\Bbb R)$

Is there a total ordering on $\mathcal P(\Bbb R)$, the set of all subsets of $\Bbb R$, such that the set of countable subsets is dense in it? (Given a total ordering $(X,>)$, a set $A\subseteq X$ ...
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1answer
55 views

The cardinality of the integers is divisible by all prime numbers?

In this question Parity of members in a group I defined even members of a group $G$ as all members $b \in G : b \neq a^ca^{c+1}$ where $a \in G$ and $c \in \mathbb{N}$ . This follows from the fact ...
5
votes
2answers
68 views

prove that if X is a countable set of lines in the plane then the union of all lines in X can't cover the plane

here's my try: Let $X$ be a countable set of lines in the plane. the cardinality of the set of all lines in the plane with a slope between $0$ and $2\pi$ is $\aleph$ so there must be some line in the ...
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0answers
21 views

Prove that if $|X|=\aleph _0$ then there exist a family of sets, $\mathcal{F}$, of subsets of $X$, s.t $|\mathcal{F}|=\aleph$ [duplicate]

Let $X$ be a set such that $|X|=\aleph _0$. I need to find a family of sets $\mathcal{F}$, of subsets of $X$ such that $|\mathcal{F}|=|\mathbb{R}|$. I saw a couple of examples of Specific X but I ...
0
votes
2answers
48 views

Prove that the set of all periodic sequences (from some index) of natural numbers is countable

This exercise is from my course textbook and comes with a bunch of other exercises which practice the theorem that countable union of countable sets is countable. So I started by notating for every ...