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
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4answers
42 views

A surjective map $S \to T$ implies $|S| \geq |T|$

Problem: Suppose that there is a function mapping $S$ onto $T$. Show that $\operatorname{Card}(S)\ge\operatorname{Card}(T)$ Issue: I can't seem to find a reason why this follows. If $S$ maps ...
1
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1answer
20 views

Possible cardinalities of the equivalence partitioning

Let $\sim$ denote a relation in $\mathbb{R}$ as follows: $x \sim y \iff d(x,y) \in \mathbb{Q} $ ($d(x,y)$ is the distance between $x$ and $y$) Determine the possible cardinalities of the equivalence ...
2
votes
0answers
28 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
1
vote
2answers
63 views

Cardinality of all rational points in $R^3$

Question: Find the cardinality of the set of all points in $R^3$ all of whose coordinates are rational, and justify the answer. Idea: Call the set of all points in $R^3$ all of whose coordinates are ...
2
votes
1answer
39 views

cardinality of the set of points with one irrational coordinate, and one rational.

Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer. My thoughts so far. We know that $\mathbb Q$, the set of all ...
1
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1answer
30 views

Cardinality of the set of minimal sets of a collection with cardinality $\aleph_0$

Let $X$ be a set and $F$ be a collection of subsets of $X$ such that $\vert F \vert = \aleph_0$ and let $F^*$ be the smallest collection of subsets of $X$ closed under intersection and complement ...
10
votes
0answers
95 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to ...
0
votes
1answer
19 views

Cardinality Proof Problem

Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A. I know that there is only an injection from A to B. I'm having trouble showing that ...
0
votes
2answers
33 views

size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
0
votes
1answer
20 views

Prove that $\{X \in P(Z)| X \text{ is finite}\}$ is enumerable. [duplicate]

I am not sure how to approach this problem. if you could help it would be great.
1
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1answer
32 views

How to proof or deproof that 2 amounts have the same cardinality?

I am new to cardinality proofs and so far I can't even understand exactly what I have to do here. Can anyone enlighten me? The task is the following: Let $M = \{ n \in \mathbb N^+ \mid n \mod 3 = 0 ...
6
votes
2answers
163 views

Prob 8, Sec 7 in Munkres' TOPOLOGY 2nd ed: How do we show these sets have the same cardinality?

Here's Prob. 8. Sec. 7 in Topology by James R. Munkres, 2nd edition: Let $X$ denote the two element set $\{0,1\}$; let $\mathscr{B}$ be the set of countable subsets of $X^{\omega}$. Show that ...
0
votes
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 $(0,1)$.
3
votes
3answers
48 views

Finding the cardinalty of a subset of $\mathcal{P}(\mathbb{N}) $

I'm trying to find the cardinality of a certain set and I'm stuck. The problem is, we haven't learned about cardinality nor about any of its rules and equalities. We are asked to find a set whose ...
1
vote
1answer
67 views

Prob 6, Sec 7 in Munkres' TOPOLOGY, 2nd ed: The existence of an injection of a superset into the set means the sets have the same cardinality?

Let $A$ and $B$ be two sets such that $B \subset A$ and there is an injection $f \colon A \to B$. Then how to show that $A$ and $B$ have the same cardinality? Munkres' Hint: We define $A_1 \colon= ...
0
votes
4answers
78 views

How to create a bijection between $(0,1)$ and $(0, \infty)$?

I don't understand how to do this. The tip I have for the question is to first find a bijection between $(0,1)$ and $(1,\infty)$.
1
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1answer
242 views

Which of the following statements are true on countable sets

Show that the numbers of the form $\sum_{k=1}^{\infty} \frac{a_j}{3^k}$ , where $a_j = 0$ or $a_j = 1$ is countable . If $A = \cap_i^n A_1$ is countably infinite, then atleat one $A_i$ is counntable. ...
0
votes
1answer
47 views

The cardinal number [closed]

Let $c$ be the cardinal number of $[0,1]$, i.e. $|[0,1]|=c$. Notice that $|A|\cdot|B| = |A\times B|$ and $|\mathbb{R}| = c$. Prove that $c\cdot c=c$. Don't use $ab=\max\{a,b\}$ where $a,b$ are ...
2
votes
1answer
74 views

Show that if X is an uncountable set and Y is a countable set then X $\bigcup$ Y has the same cardinality as X [closed]

Show that if X is an uncountable set and Y is a countable set then X $\bigcup$ Y has the same cardinality as X. Is this okay? $$$$ (X $\bigcup$ Y) = X + Y - (X $\bigcap $Y) $$$$ (X $\bigcup$ Y) + (X ...
1
vote
1answer
24 views

Equinumerousity of two sets

Given two infinite sets $A$ and $B$, I'm asked to show that the two sets $\mathcal{P}(B)^A$ and $\mathcal{P}(A)^B$ are equipotent. I proved it by showing that those two sets have the same cardinality ...
1
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0answers
17 views

Multisets and cardinality

Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. ...
1
vote
0answers
14 views

Show cardinality problem with natural evens, all naturals and all integers

I know how to show the cardinality between 2 of the 3 could work. For example, you can establish a bijection with integers and all naturals through $\mathbb{N}$: s(n) = $\sum_{k =1}^n1$ and ...
-2
votes
1answer
38 views

Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
0
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0answers
28 views

Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
1
vote
1answer
28 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
1
vote
1answer
51 views

definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
7
votes
2answers
543 views

Cardinality of the Euclidean topology and the axiom of choice

It is relatively straightforward to prove that the Euclidean topology has the same cardinality as the space itself. I have sketched a proof below. The proof seems to rely quite heavily on the axiom of ...
3
votes
1answer
60 views

Trying to understand the slick proof about the dual space

In this famous MO question, a beautiful proof is given of the fact $V\cong V^\ast\iff V$ is finite dimensional. I'm trying to go through it and I'm having some trouble. First of all, I know the in ...
1
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0answers
52 views

Examples of uncountable fields of characteristic $p$?

Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an ...
1
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2answers
71 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
0
votes
1answer
26 views

Cardinality of a metric space

Let $X$ be an infinite set .For any two metrics $d_1,d_2 $ on $X$ the identity map $i:(X,d_1)\to (X,d_2)$ is continuous. Prove that $X$ is always countable. I am not getting how to start this ...
-1
votes
1answer
43 views

If a countable union of sets has card $\mathfrak{c}$, prove at least one of them has card $\mathfrak{c}$ [duplicate]

If $A=\bigcup_{n=1}^{\infty}A_n$ and $A$ has cardinality $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinal of the continuum, prove that at least one of the $A_n$ has cardinality $\mathfrak{c}$.
1
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1answer
41 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that ...
0
votes
2answers
52 views

Show that $\leq$ is transitive over the cardinality of sets

$A, B, C$ are sets. I want to show $|A|\leq|B| \ \text{and} \ |B|\leq|C| \Longrightarrow |A|\leq|C|$. I am confused as to how I would approach this, because the sets in this problem can be either ...
6
votes
4answers
2k views

Taking away infinitely many elements infinitely many times [duplicate]

This is a somewhat hand wavy question but I'm not sure how to ask it more precisely. If we have a countably infinite sequence (or set), can we take away infinitely many elements from the sequence ...
1
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1answer
19 views

cardinal of the union of an increasing sequence

Let $A_n\subset \mathbb{R}$ for all $n$. If $\vert A_n\vert=C$ for all $n$ where $C$ is the cardinality of the real numbers and $A_n\subset A_{n+1}$. then Is it true that $\vert \bigcup_n ...
0
votes
2answers
59 views

Show that the set is not countable

To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example: ...
1
vote
1answer
25 views

cardinality of the set of junctions intervals

Let P$=\{(a,b) : a,b \in \mathbb{R}\}$ and Q$=\{$ countable unions of elements of $P\}$. I 'm interested in knowing the cardinality of Q. I denote with $C$ if the cardinal real , then $C \leq \vert ...
5
votes
1answer
59 views

Cardinals and generic extensions

Let us consider a model of ZFC $M$, a forcing $\mathbb{P}\in M$. If $G$ is a $\mathbb{P}$-generic filter over $M$ we can construct the generic extension $M[G]$ of $M$. Given any cardinal number ...
4
votes
1answer
89 views

Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
0
votes
4answers
180 views

Proving a set is Uncountable or Countable Using Cantor's Diagonalization Proof Method

I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list. I have ...
3
votes
3answers
99 views

Example 2, Sec 7 in Munkres' TOPOLOGY 2nd ed: How to show that $\mathbb{Z}_+ \times \mathbb{Z}_+$ is countable?

We need to show that there is a bijection $h \colon \mathbb{Z}_+ \times \mathbb{Z}_+ \to \mathbb{Z}_+$. For this purpose, Munkres define a subset $A$ of $\mathbb{Z}_+ \times \mathbb{Z}_+$ as follows: ...
3
votes
2answers
126 views

Minimal foundations for Cardinal Arithmetic

I would like to develop a theory of cardinal numbers that relies on as weak a basis as possible. Therefore, I would like to know if there is a way to even define a cardinal number for every set ...
0
votes
1answer
27 views

Cardinality of the set of functions which holds the Equality

Let $f$ be a function from $\{1,2,3, \dots ,10 \}$ to $\mathbb R$ such that $$\bigg( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\bigg)^2 = \bigg( \sum_{i=1}^{10} |f(i)|^2 \bigg) \bigg(\sum_{i=1}^{10} ...
1
vote
3answers
87 views

Let S be the set of all real numbers in the interval (0; 1) whose decimal expansions contain only 0's, 4's and 8's. Prove that S is uncountable.

I just prove $$S_1=\{0.4,0.8\}$$ $$S_2=\{0.04,0.08,0.44,0.48,0.84,0.88\}$$ ... So I can calculate the number of elements in $S_n = 2 \times 3^{n-1}$ I just prove it is countably infinite.
0
votes
2answers
28 views

prove $|Q|$ = $|Q*Q|$

I have this problem: Prove that $|Q|$ = $|Q*Q|$ I know that $Q$ is countably infinite But then how can I prove that $|Q*Q|$ is countably infinite? Thanks you!
0
votes
2answers
23 views

Cardinality proof verification

Problem: Let $C \subset (0,1)$ be the set of all numbers whose unique decimal representation contains the number seven. Show that the number of elements in $C$ must be the same as the number of ...
0
votes
0answers
19 views

Show that |℘(ℕ)| ≤ |Maps(ℕ,ℕ)| [duplicate]

I want to show that there exists an injective map from ℘(ℕ) to Maps(ℕ,ℕ) but I have not an idea of how to go about this. I am thoroughly confused every time I see something like Maps(ℕ,ℕ). Please ...
1
vote
1answer
66 views

In how many ways is my crocodile shaped like itself?

Here is a topologically accurate photograph of my crocodile: In particular, topologically he is a solid sphere. He can be described as being shaped like himself, but that is vague. There are so ...
0
votes
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 ...