All Questions
Tagged with cardinals real-analysis
167
questions
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational
numbers.
$f$ is strictly increasing in both
arguments.
Can $f$ be one-to-one?
This question is related to many ...
- 3,824
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0
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43
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How does measure theory deal with higher cardinalities?
The second part of the definition of a sigma-algebra is that countable unions of measurable sets are measurable. The second property of a measure is that the measure of countable unions of measurable ...
- 11
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55
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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 ...
- 1,364
1
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1
answer
186
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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) \...
- 377
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81
views
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
2
votes
0
answers
77
views
Correctness of Proof and Use of Axiom of Choice (Analysis I by Terence Tao)
I've skipped over some of the references in my proof for brevity. The following is an exercise from Terence Tao's Analysis I, specifically section 8.1. on countablity. ("countable" here ...
- 135
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1
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148
views
Why isnt $|\mathbb{R}| = |\mathbb{N}|$?
Question: To show that 2 sets have the same cardinality, there needs to be atleast one bijective mapping between them. So given the below proof of a bijective mapping below, why can't we say that $\...
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33
views
Uncountable union of chain of subsets of R
For each $t \in R$, let $E_t$ be a subset of $R$. Suppose that if $s<t$ then $E_s$ is a proper subset of $E_t$.
$$\bigcup_{t\in R} E_t$$
Is countable.
How?
I see that the union runs over R which is ...
- 389
1
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2
answers
87
views
Classification of Open Subsets of $\mathbb{R}^2$
Is there some nice-ish characterization of open subsets of $\mathbb{R}^2$? For example, open subsets of $\mathbb{R}$ can be represented as the countable union of disjoint open intervals, so they can ...
- 491
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1
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84
views
Non-existence of a surjection
Let $A,B,$ and $C$ be sets such that there is no surjection from $A$ to $B,$ and there is no surjection from $B$ to $C.$ Show that there is no surjection from $A$ to $C.$
When $A,B,$ and $C$ are ...
- 2,145
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answers
119
views
Collection of functions $f: \mathbb{R} \to \mathbb{R}$ is uncountable.
Let $F$ be the collection of all functions $f: \mathbb{R} \to \mathbb{R}$. Prove that $F$ is uncountable.
Proof:
Let $r \in \mathbb{R} $. Then the function $f: x \to x^r$ is in $F$. Since the reals ...
- 11
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votes
1
answer
219
views
What is the cardinality of $\mathbb N \times \mathbb R$? [duplicate]
I was wondering what is the cardinality of $\mathbb{N} \times \mathbb{R}$. My guess is that it is the same as $\# \mathbb{R}=c$ but I haven't been able to prove it.
I was thinking that one can easily ...
- 479
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423
views
How many 2D objects fit into a 3D object?
Hoe many times can you stack 2D objects before it becomes 3D?
I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional ...
- 11
1
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2
answers
968
views
Are most real functions non-linear?
I made an observation that for two finite sets $A$, $B$ that most $R \subseteq A \times B$ where $R$ is a function also 'appear to be' non-linear. It got me wondering if this is true in the highly ...
- 1,876
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96
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The number of Darboux functions [closed]
What is the number of real-valued functions of a real variable that have the Darboux property? I know it's at least continuum, because it's a broader class than continuous functions, whose number is ...
