All Questions
Tagged with cardinals solution-verification
152
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Irrational numbers Cardinality.
The real numbers, $\mathbb{R}$, are uncountable and the rational numbers, $\mathbb{Q}$, are countable. We can write $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$. Since $\mathbb{Q}$ ...
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The cardinality of specific set $A\subset \mathbb{N}^{\mathbb{N}}$
Let $A$ be a set of total functions from the naturals to the naturals
such that for every $f\in A$ there is a finite set $B_f\subset \mathbb{N}$ , such that for every $x\notin B_f$ , $f(x+1)=f(x)+1$.
...
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Prove $C ∼ P(P(\mathbb{N}))$ when $C$ is defined as the set of all $S$ s.t $(z − m, z + m) ∩ S = ∅ $ for every $z∈\mathbb{Z}$, $m∈\mathbb{R}$
First, I know that there is a very similar question - The cardinality of all sets $A$ such that $\forall \ z\in \mathbb{Z} \ , (z-k,z+k)\ \cap A=\emptyset : 0<k<0.5$ , but here I want to the ...
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Fixing my gripe with the common proof for showing that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}|$
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
Showing that $f:(-1, 1) \...
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81
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Show that $\mathcal{O}$ the set of all open sets in $\mathbb{R}$ has the same cardinality as $\mathbb{R}$
I've seen the post from here Prove that the family of open sets in $\mathbb{R}$
has cardinality equal to $2^{\aleph_0}$
This post is somewhat complex for me, and I turned it to the question as my ...
- 1,364
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56
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Cardinality of set of all rectangles on a plane
The set of rectangles in $\mathbb{R}^2$ with sides parallel to axes and bottom-left corner in origin can be described as:$$\mathcal{R}=\{[0,a]\times[0,b]:a,b\in\mathbb{R}\}$$
There is bijection $\...
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57
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Bound for the cardinality of the model of a set of formulas
I don't know if the proof of the theorem below is correct. The goal is to do the proof from the beginning, without using compactness or Löwenheim-Skolem. Only the Tarski-Vaught test (maybe).
We are ...
- 414
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Is $\operatorname{cf}(2^{<\kappa})=\operatorname{cf}(\kappa)$?
As the title says, I am wondering whether $\operatorname{cf}(2^{<\kappa})=\operatorname{cf}(\kappa)$ where $\kappa$ is an infinite cardinal. I believe I have a proof: Define $f:\kappa\to 2^{<\...
- 598
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Is Lagrange group theorem still valid for infinite groups?
So it is a well know result that the quotient space $G/\cal R$ corresponding to right congruent relation $\cal R$ is equipotent to the quotient space $G/\cal L$ corresponding to the left congruent ...
- 4,906
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Proposition: The union of two finite sets is finite. (proving without induction)
Is my proof correct?
Notations:
$\mathbb{N}$:=$\{1,2,3,..
\}$ = The set of all natural numbers.
$J_q$:=$\{ y : 1\leq y \leq q, \text{ for some } q \in \mathbb{N}\}$ = The set of first $q$ natural ...
- 23
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How to prove that Q \ Z is countable
The question is asking to prove that Q \ Z is countable. We are allowed to use the fact that Q and N have the same cardinality.
My current idea is to show that there is a bijection between the set Q \ ...
- 51
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54
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Prove or disprove that if $f:X\longrightarrow Y$ is a perfect map then $w(Y)\le w(X)$
Let be $f$ a perfect map from a space $X$ to a space $Y$ so that I am trying to prove or disprove if the inequality
$$
\begin{equation}\tag{1}\label{1}w(Y)\le w(X)\end{equation}
$$
holds, where $w$ is ...
- 4,906
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41
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Cardinality of the set of internal sets
Non standard analysis relies on using sets that have the same properties as sets of real numbers to transfer many good theorems over. Every set $S\in P(\Bbb R)$ has a unique extension $S^* \in P(\Bbb ...
- 1,207
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[FEEDBACK]: Proving that the union of any two infinite countable sets is countable [duplicate]
I made the following proof for an excercise regarding the union of two countable infinite sets, namely, that their union is countable. It would be of great help if anyone could give me some feedback ...
- 187
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What's wrong with this proof that $|\omega_1|\ge2^{\aleph_0}$?
According to this question, the size of $S_{\mathbb N}$, the set of all bijections on $\mathbb N$, is at least $2^{\aleph_0}$. So $|S_{\mathbb N}| \ge 2^{\aleph_0}$
For each bijection $f:\mathbb N\...
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