All Questions
Tagged with cardinals elementary-set-theory
1,403
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proving the set of natural numbers is infinite (Tao Ex 2.6.3)
Tao's Analysis I 4th ed has the following exercise 3.6.3:
Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
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Is the cardinality of $\varnothing$ undefined?
It is intuitive that the cardinality of the empty set is $0$.
However we are asked to demonstrate this using given definitions/axioms in Tao Analysis I 4th ed ex 3.6.2.
My question arises as I think ...
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How should I should prove $\mathbb{R}\sim\{0,1\}^{ \mathbb{N}}$ [duplicate]
I've seen some argument about the binary representation, but I think it is not accurate because under some extreme cases, the rounding or bit constraint would results distinct reals also have the same ...
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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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Finite cardinals raised to the power of an infinite cardinal
I am trying to prove the fact that if $a$ and $b$ are finite cardinals, and $c$ is an infinite cardinal, then $a^c = b^c$. I am able to prove this fact by using $d \cdot d = d$ for all infinite ...
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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 ...
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Set theory - functions: Given that $|A|=|B|$, how to prove that $|A^C|=|B^C|$?
Given that $|A|=|B|$ (the cardinality of set $A$ is equal to the cardinality of $B$).
How can I prove that $|A^C|=|B^C|$ (the cardinality of the set of all functions from $C \longrightarrow A$ is ...
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How the comparison of the cardinalities of sets affects the cardinalities of their powersets [duplicate]
In my question, I denote by $|\cdot|$ the cardinality of any set. Moreover, if $f: X \to Y$, we denote by $\mathcal{P}f$ its direct image, i.e. $\mathcal{P}f(A)=\{f(a) : a \in A\}$. Let $X,Y$ be two ...
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Cardinality of Infinite Sets in a Cartesian Product
I'm trying to show that $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ and $\mathbb{R}^2$ have the same cardinality.
It has been given to me that for any finite interval (open or closed), it has the same ...
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Constructing a bijection between $\mathbb{N}^2$ and $\{(x, y) \in \mathbb{N}^2: x \geq y\}$
Bijection between $\mathbb{N}^2$ and $\{(x, y) \in \mathbb{N}^2: x \geq y\}$.
My idea is to walk diagonally on $\mathbb{N}^2$, starting from $(0, 0)$, then to $(1, 0)$, $(0, 1)$ etc. Hopefully I can ...
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How to define the sum of cardinals in the definition of an inaccessible cardinal?
I'm reading the Wikipedia definition of an inaccessible cardinal and I'm trying to understand it.
On Wikipedia, a (strongly) inaccessible cardinal $\kappa$ is defined in the following way:
$\kappa$ ...
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The existence of a function on $\kappa$
Let $\kappa>\omega$ be a regular cardinal. Let $C\subseteq\kappa$ be a club. Prove that there exists a function $f:\kappa\rightarrow\kappa$ such that $C_f=\{0<\alpha<\kappa:\forall\xi<\...
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Prove that these sets are stationary.
Define Lim($\omega_1$)={$\delta<\omega_1\ :\ \delta$ is a limit ordinal}. Assume that $\left<A_\alpha:\alpha\in\text{Lim}(\omega_1)\right>$ is a sequence satisfying the following:
(1) $\...
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To show the cardinality of $\mathbb{R}$ and $\mathbb{R}^\mathbb{N}$ are same [duplicate]
I want to show that the cardinality of the countable infinite product of the set of real numbers $\mathbb{R}$ is same as the cardinality of $\mathbb{R}$. I am trying to find a bijection from $\mathbb{...
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Proving proper inequality of cardinality of sets is transitive
I need to prove the following:
If $A,B,C$ are sets (finite or infinite), such that $|A|< |B|$, and $|B| < |C|$ then $|A| < |C|$.
My proof:
Let $f:A \to B$ be in injective, since $|A| \le |B|$,...
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