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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4
votes
2answers
231 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?
3
votes
2answers
36 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 ...
1
vote
2answers
29 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?
1
vote
2answers
64 views

König's theorem (set theory) implication

How does König's theorem imply $\quad\aleph_{\omega} \neq \beth_1$?
3
votes
7answers
413 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?
-1
votes
3answers
310 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
votes
1answer
84 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
votes
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
votes
1answer
128 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
77 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
61 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
557 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
votes
1answer
85 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
votes
0answers
75 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
votes
1answer
57 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
votes
1answer
39 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
vote
1answer
46 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
votes
0answers
38 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
votes
1answer
29 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
vote
1answer
58 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
votes
2answers
46 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 ...
-3
votes
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
75 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 ...
4
votes
4answers
294 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
213 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
89 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
54 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
83 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$\} $?
-1
votes
1answer
115 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
119 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
80 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
votes
3answers
197 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 $A\subseteq X$ ...
-2
votes
1answer
63 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
152 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 ...
1
vote
0answers
22 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
55 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 ...
3
votes
1answer
40 views

Does every set have choice sequences as long as the original set?

Given a set $X$, we say that $X$ has choice sequences of length $|I|$, denoted $CS(|X|,|I|)$, if for any $f:I\to{\cal P}(X)\setminus\{\emptyset\}$ there is a function $g:I\to X$ such that $g(x)\in ...
3
votes
2answers
76 views

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$ This is from Hodges' A Shorter Model Theory. My idea is to take ...
1
vote
1answer
39 views

proving $|X|<|Y|$, $|Y|<|Z| \Longrightarrow |X|<|Z|$ without CSB

how to prove that if $|X|<|Y|$, $|Y|<|Z|$ then $|X|<|Z|$ without CSB theorem? it is immediate that $|X|\leq |Z|$ so I tried to assume that $|X|=|Z|$ and reach a contradiction but so far I ...
1
vote
2answers
48 views

Characterizing uncountable connected topological spaces

We know that if $X$ is a connected metric space with more than one point , then $X$ is uncountable ; can we characterize those connected topological spaces for which more than one point implies ...
3
votes
1answer
50 views

How to prove that if $A$ is infinite and $B$ is finite, then $|A\cup B|=|A|$?

I'm studying logic and unfortunately, I'm a newbie at this, so I don't see the stuff everyone sees at the moment. I want to solve following exercise, but get nowhere: Let $A$ be an infinite set ...
3
votes
0answers
46 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
3
votes
1answer
26 views

Let $\Gamma$ be a $\kappa$-based monotone operator where $\kappa$ is regular. Then the closure ordinal of $\Gamma$ is $\kappa$.

A monotone operator $\Gamma: \mathcal{P}(A) \to \mathcal{P}(A)$ is an operator such that, if $X \subseteq Y \subseteq A$, then $\Gamma(X) \subseteq \Gamma(Y)$. A monotone operator is $\kappa$-based ...
1
vote
1answer
71 views

How to calculate the cardinality of the complement of two countable sets of reals?

Let $A,B\subseteq\Bbb R$ be countable sets. Denote by $A'$ and $B'$ the complements (in $\Bbb R$) of $A$ and $B$ respectively. What is the cardinality of $C=A'\cap B'$? I cant figure this ...
3
votes
1answer
69 views

Anatomy of $\mathcal P(\mathbb{N})$

How many proper subsets of $\mathcal P(\mathbb{N})$ there is that have cardinality of $2^{|\mathbb{N}|}$ ?
3
votes
1answer
222 views

Counting quantification and the cardinality of a set

A counting quantifier is a quantifier that denotes how many elements satisfy a predicate. I will use the notation $C_n x P[x]$ to denote that there are $n$ elements that satisfy $P$. I was thinking ...