0
votes
0answers
29 views

The limit of sequence of set.

When dealing with sequence of numbers, if $a_n\rightarrow a$, and for each $a_n$, we have $a_{n,j}\rightarrow a_n$, I think there exists a subsequence $a_{n,n_j}\rightarrow a$. What if we replace ...
0
votes
2answers
36 views

Is there a straight-forward, “magic bullet” style way of showing $(\overline{X^\mathsf{c}})^\mathsf{c} = X^{\circ}$?

I would like to rigorously show that $(\overline{X^\mathsf{c}})^\mathsf{c} = X^{\circ}$, that is, the complement of the closure of the complement of X equals the interior of X. I am TAing a class ...
1
vote
0answers
26 views

The product of two rational Dedekind cuts

If $a,b\in \mathbb{Q}$ and $C_a$ and $C_b$ are both positive rational Dedekind cuts then $C_a\cdot C_b=C_{a\cdot b}$. First of all this is my definition of product: Let $r,s$ Dedekind cuts such ...
1
vote
2answers
35 views

How show $\mathbb N \cong \mathbb Q$ using Cantor pairing?

According to this: http://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor_pairing_function, we can show that $\mathbb N\times\mathbb N\cong\mathbb N$. But as for $\mathbb Q$, this is not the ...
-6
votes
1answer
79 views

Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
1
vote
1answer
41 views

What is $\bigcup_{r\in(0,1)}[0,r]$?

Question: for any real number $r$, let $C_r$ be the closed interval $[0,r]$. Let $J$ be the open interval $(0,1)$. what is $\bigcup_{j\in J} C_j$? So far I have attempted a double inclusion proof to ...
0
votes
1answer
47 views

Cardinality of a set of Continuous and Real Functions in the interval $[0,1]$

my question reads as: Let $\mathcal R[0,1]$ denote the set of all real-valued functions from $[0,1]$ to $\mathbb R$ and let $\mathcal C[0,1]$ denote the set of continuous functions on $[0,1]$. ...
0
votes
1answer
34 views

My Proof for the Cardinality of a Particular Binary Distribution

my question reads as follows: I have constructed a proof and am concerned about 2 things: 1) The validity of my proof. 2) The construction of my proof. I am asking for someone to read through ...
0
votes
4answers
101 views

Why there is not the next real number?

We can't say what is the just next real number (or rational or irrational number) of a given real number (or rational or irrational number respectively), what is the actual fundamental reason behind ...
4
votes
2answers
95 views

Cantor Set Uncountability [duplicate]

The Cantor set is closed in $[0,1]$ and so its complement in $[0,1]$ should be a countable union of open intervals. Furthermore, every open set containing a point in the Cantor set contains a point ...
1
vote
1answer
63 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
0
votes
1answer
45 views

Show that the set is a singleton

Let $f$ be convex, differential function. Consider the set $$X=\left\{x\in \underset{x}{\text{argmin}} f(x):\; \|x\|\leq \|y\|,\;\forall y\in \underset{x}{\text{argmin}} f(x)\right\}$$ Prove that this ...
0
votes
1answer
19 views

uncountably many numbers with lower bounded difference?

Does there exist an uncountable set U of reals s.t. $\forall a,b \in U\exists k>0$ giving $|a-b|>k$? This is impossible right? Because if not, then $\{(a,b)_{\lambda}\}$ will be uncountably ...
0
votes
1answer
41 views

Real numbers in interval notation

If I want to denote the set $\mathbb{R}$ in $[a,b]$ interval notation, is it correct to say: $ (-\infty, \infty)? $
0
votes
0answers
28 views

What is the symbol $B_a$ mean in set notation

What is the symbol $B_a$ mean in set notation? $$\{Ba∣a∈A \text{and } B_a \text{ is the set of children of a}\}$$ I have seen the above used as an answer here. But I am not quite clear on what is ...
0
votes
1answer
41 views

Check my proof of the “Boundedness theorem”

Theorem: Let $f$ be continuous on a closed interval $[a, b]$. Then f is bounded on $[a, b]$. Proof (sketch): Suppose $f$ is unbounded. Let's define the set $N$ containing those $x$ for which $f$ is ...
3
votes
0answers
91 views

My proof of Bolzano's theorem

Before I read the proof of Bolzano's theorem from my Calculus book, I've tried to prove it myself. I will use the following lemma and the least upper bound axiom. [Lemma: Sign-preserving property of ...
0
votes
2answers
34 views

Can I define sets as an infinite process?

Can I define sets this way in real analysis / set theory? I mean defining sets in kind of an infinite process and then taking their supremum or infinum. Let $S = \{1\}$. And for every element $x$ in ...
0
votes
1answer
96 views

A basic property of the Lebesgue outer measure

If $G$ is a measurable set and satisfies $m^*(G)<\infty$, then for all $\varepsilon>0$ there exists a closed set $F\subset G$ such that $m^*(F)>m^*(G)-\varepsilon$ Edit: I know that: ...
0
votes
5answers
135 views

How to prove that $\mathbb{Q}$ ( the rationals) is a countable set

I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it ...
0
votes
1answer
41 views

Is it possible to form a bijection from N to Z if 0 is not an element of N?

I'm trying to show a bijection from $\mathbb{N}$ to $\mathbb{Z}$ for this assignment, which isn't terribly difficult however 0 is not included in $\mathbb{N}$ as defined by this book i'm working in... ...
2
votes
2answers
110 views

Why does $\mathbb{R}$ have the same cardinality as $\mathcal{P}(\mathbb{N})$? [duplicate]

I recently read a fact that surprised me: The power set of the natural numbers is equinumerous with $\mathbb{R}$. In other words, $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}|$. I don't intuitively see ...
1
vote
1answer
62 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
-1
votes
2answers
81 views

Proving that a certain continuous function is surjective.

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $|f(x)-f(y)|≥|x-y| ,\forall x,y \in \mathbb R $ , then how do we prove that $f$ is surjective ?
1
vote
1answer
58 views

Are there any known uncountable transfinite increasing sequences of real numbers?

The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ..., $ up to (and possibly ...
4
votes
2answers
184 views

Exercise Real Analysis

I'm having trouble to understand the following exercise I would appreciate any help? Let $A,B,C$ be sets such that $A\subseteq B\subseteq C$ and let $f:C\rightarrow A$ be an injective map. Define ...
-1
votes
3answers
363 views

Exponentiation as repeated Cartesian products or repeated multiplication?

In set theory, if $A$ and $B$ are sets, then their Cartesian product is defined to be $A\times B$ such that: $\forall x,y: [(x,y)\in A\times B \iff x\in A \land y\in B]$ Exponentiation (as repeated ...
0
votes
2answers
40 views

Closed subset of R^2

Show that the set A = {(x,y): $x^3$ $>=$ $y^5$} is closed as a subset of $R^2$. I defined a closed set as a set whose complement is open. So the complement of the above set is {(x,y): $x^3$ ...
0
votes
2answers
51 views

How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there …

I am stuck on the following problem when I was trying to solve an entrance exam paper: How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there with the property that $\, ...
2
votes
1answer
166 views

Explicit Bijection from $\mathbb{N}$ to $\mathbb{Q}^+$.

Reading Bartle and Sherbert's intro to Real Analysis and going over denumerable sets. I know because of a diagonal procedure that this bijection exists, but I've been trying to find an explicit ...
0
votes
1answer
33 views

Transform this expression in something useful for me

I have $A \subset \Omega$ and $E \subset \Omega$. Now I have $A\cap E^C$. But I do not want to work with the complement, I am rather looking for an expression that somehow contains $A \cap E$. Does ...
1
vote
1answer
45 views

Baire Category Theorem: What should we really prove there?

I am reading about the Baire Category Theorem in Jech's book on set theory. 4.8: Baire Category Theorem: Let $D_0,D_1,\dots,D_n,\dots$, $n \in \mathbb{N}$, be open dense subset of $\mathbb{R}$. Then ...
0
votes
1answer
21 views

Measure and empty set second version

If A is a nonempty open set, the measure of A is not 0 ...first open set is uncountable, I pick up one interval irrational intersect [ 0,1 ] with measure 1 ...however this is not open...I am wondering ...
1
vote
1answer
141 views

Is there a bijective map from the open interval $(0,1)$ to $\mathbb{R}^2$?

I couldn't find a bijective map from the open interval $(0,1)$ to $\mathbb{R}^2$. Is there any example?
-2
votes
1answer
65 views

Cardinality of “x−y∈Q”-equivalence class of 1/2 √ [duplicate]

For x,y∈I:=[0,1] define the relation on I as x−y∈Q. How big (using cardinal number) is the cardinality of the equivalence class [1/√2]? I have tried to solve it by finding the equivalence class but ...
1
vote
3answers
115 views

Real Numbers as Well Defined Sets

For every construction of the reals, we define a real number to be some kind of set of rational numbers (such as cuts or sequences). However, the number of symbols we have to formulate the ...
1
vote
0answers
47 views

Set Union Notation $A_{\mathbb{N}}$ and question on proof

I was doing some rudimentary set theory problems to keep sharp and I came across a problem: Let $A_n=\{(n+1)k|k \in \mathbb{N}\}.$ Find $\bigcup \{A_n|n \in \mathbb{N}\}$ So, is it appropriate to ...
4
votes
1answer
617 views

Prove that ℝ and the interval (0, infinity) have the same cardinality.

Prove that Real Numbers and the interval (0,∞) have the same cardinality. Attempt: Consider the function f(x) = e^x. The domain of this function is all real numbers. The range of this function ...
-1
votes
2answers
108 views

Cardinality of “$x-y\in\Bbb Q$”-equivalence class of $1/\sqrt2$

For $x,y\in I:=[0,1]$ define the relation on $I$ as $x-y\in \Bbb Q$. How big (using cardinal number) is the cardinality of the equivalence class $[1/\sqrt2]$? I have tried to solve it by ...
1
vote
0answers
24 views

How does one “separate” the cartesian product properly?

Say, $\delta>0$, $X$ and $Y$ are metric spaces, $(x_0,y_0)\in X \times Y $, and there is some property $P$ such that $$\forall (x,y) \in X \times Y: \ \ \ \ d \Big( (x_0,y_0), (x,y) \Big) < ...
2
votes
1answer
46 views

proving for cantor set: $\mathcal C=\frac13\mathcal C\cup\left(\frac23+\frac13\mathcal C\right)$

Let $\mathcal C$ be the Cantor set. Then $$\mathcal C=\frac13\mathcal C\cup\left(\frac23+\frac13\mathcal C\right)$$ with $\frac13\mathcal C:=\{\frac13x:x\in\mathcal C\}$ and $\frac23+\frac12\mathcal ...
1
vote
1answer
29 views

Bijectivity of set sequences

I've got this homework problem to prove in my introductory analysis course ... and right now, I really have no idea how to even go about that (and as such, don't really know the right questions to ...
1
vote
1answer
210 views

Sigma field generated by Borel sets is the same as sigma field generated by intervals

Let $\mathcal{R} = \{ B_1 \times B_2 : B_1,B_2 \in \mathcal{B} \} $ where $\mathcal{B}$ is the sigma field of Borel sets. Let $\mathcal{I} = \{ I_1 \times I_2 : I_1,I_2 \; \; \text{are intervals} \} ...
2
votes
2answers
91 views

Zero, the Additive Identity, as the Multiplicative Annihilator

In the structures I have encountered so far, I have always seen a zero, which is usually defined as the additive identity. For example: $\exists 0 \in \mathbb{Z}$ s.t. $\forall a \in \mathbb{Z}, a ...
1
vote
3answers
201 views

The set of real numbers

I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these ...
0
votes
2answers
132 views

$\mathbf{Q}$ is not a countable intersection of open sets $\rightarrow$?

I need help understanding a statement. I have been told that : $\mathbf{Q}\subset\mathbf{R}$ is not a countable intersection of open sets. In other words, ...
3
votes
2answers
436 views

A finite set always has a maximum and a minimum.

I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated.
2
votes
1answer
36 views

Inequality with the supremum

I am trying to prove the following statement for $A\subset \mathbb{R},~\epsilon>0$ with $A$ bounded above: $\sup(A)-\epsilon<a\leq\sup(A)$, for some $a \in A$ I have tried dividing it into two ...
1
vote
2answers
106 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
2
votes
1answer
86 views

Question about a proof that The Cantor set is uncountable.

I am reading a proof from a paper I found online, and it goes like this: We want to show that there exists a surjection $f$ from the cantor set $\mathfrak{C}$ to the interval $[0,1]$ then we can show ...