5
votes
0answers
80 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
1
vote
2answers
44 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
3
votes
4answers
104 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). ...
1
vote
0answers
22 views

problem with my construction of sequence of sets

This is an exercise from a real analysis book. It has many parts. Assume a) and b) are true: a) Suppose $F$ is closed and $O$ is open subset of $\mathcal{R}$ and $F \subset O$, then there is a ...
2
votes
2answers
50 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...
1
vote
0answers
19 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
1
vote
1answer
54 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
2
votes
1answer
27 views

How to express membership to at least $m$ sets in a sequence of sets.

Suppose we have a sequence of sets $(A_{n})$. Pick some positive integer $m$. How would you express the set of all points that belong to at least $m$ sets in the sequence $(A_{n})$? I tried toying ...
1
vote
1answer
43 views

Problem in set theory

Let $h:\mathbb R\to \mathbb R$ such that $\forall x\in \mathbb R\Rightarrow h(x)>0$. Prove that there exist $A\subset \mathbb R$ and $\epsilon>0$ such that $A=_c\mathbb R$ and $\forall x\in ...
6
votes
2answers
81 views

$\Gamma \subset \mathbb{R}^{+}$ is uncountable. Can we choose a sequence from $\Gamma$ of which the sum is $\infty$

If $\Gamma$ is a set of uncountably many different positive real numbers, can we choose a sequence of pairwise different positive numbers from $\Gamma$, say $\{a_n\}$, such that $\sum a_n = \infty$ ...
2
votes
2answers
19 views

How many tagged partitions of an interval are there?

A tagged partition of an interval $[a, b] \ (a, b ∈ ℝ, a < b)$ is a finite sequence $(x_i)_{i=0}^n$ in $ℝ$, where $a=x_0 < x_1 < … < x_n = b$. Consider the set of all tagged partitions of ...
0
votes
1answer
62 views

$f(A) \cap f(B) = f(A \cap B)$ if $f$ is a bijection?

I found this statement in a Topology proof - $$f(A) \cap f(B) = f(A \cap B)$$ if $f$ is a bijection I haven't come across this statement before. Is this some axiom of set theory?
2
votes
0answers
65 views

Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
0
votes
0answers
29 views

Right continuous but not left continuous function and cardinality

I have been given the following question, Let $f:\mathbb{R}\to\mathbb{R}$ be an arbitrary function and $L=\left\{x:\text{f is right continuous but not left continuous at }x \right\}$. Prove that $L$ ...
2
votes
3answers
94 views

What is the difference between a set that is countable and infinite and one that is countably infinite?

I am rereading my analysis notes and I came on this remark in the section on countability: We have proved that Q is countable, and certainly Q is not finite, because N ⊆ Q. We have not proved that Q ...
0
votes
1answer
20 views

Additive inverse of a Dedekind cut — is my definition alright?

I am struggling to understand why the Additive Inverse is typically defined as such: $$\alpha^∗ := \{x \in \mathbb Q | \exists r > 0\text{ such that }−x−r\notin \alpha\}$$ or in another form ...
3
votes
1answer
35 views

Showing that for $s,t\in\mathbb{Q}$, we have $(s+t)^*= s^* + t^*$.

I'm working through the problems of Elementary Analysis Theory of Calculus, and for some reason, this question didn't make the solutions in the back of the book. I did a thorough search on Stack ...
0
votes
2answers
34 views

Countability of the Real Number set using an infinite-dimensional array

My friend and I were talking about Cantor's Diagonal Argument, and he was asking why the Real Numbers were uncountable. He proposed the following situation: On the first axis, we put 0, 1, 2, ..., 9 ...
0
votes
2answers
29 views

Continuity of set inclusion function

Let $A_1$, $A_2$, $A_3$, $\cdots$ be a sequence of nonempty subsets of $[0,1]$. For $x \in [0,1]$, set $a_i (x) = 1$ if $x\in A_i$ and $0$ otherwise. Define $f(x) = (0.a_1 (x) a_2 (x) a_3 (x) \cdots ...
0
votes
2answers
33 views

Does $A \subset \cup_{i=1}^{\infty}U_i$ and $B \subset \cup_{n=1}^{\infty}V_n$ imply $A \times B \subset \cup_{i,n}(U_i \times V_n)$?

If $A \subset \cup_{i=1}^{\infty}U_i$ and $B \subset \cup_{n=1}^{\infty}V_n$, how can I show that $A \times B \subset \cup_{i,n}(U_i \times V_n)$? Also, what would it mean for $A \times B \subset ...
1
vote
1answer
30 views

How to embed a total ordering into the real field.

Let $(S,<_S)$ be a total ordering with $card(S)\leq card(2^{\aleph_0})$. Does there exist a subset $A$ of the real numbers such that $(A,<_A)$, being a total ordering, is isomorphic to ...
1
vote
1answer
40 views

$n$-fold compositions and countability

Consider $\phi:\;(0,1)\to (0,1)$ satisfying $\phi(x)<x$ and let $\phi^n(x)$ denote $\phi(\phi(\cdots \phi(x)\cdots )$ iterated $n$ times. Then $\phi^n(x)$ converges for each $x\in (0,1)$. Hence ...
3
votes
0answers
51 views

When can we have $(A+B)\cap C=A\cap C+B\cap C$?

With $A+B=\{a+b:a\in A, b\in B\}$ and any non-empty sets A,B,C. When can we have $(A+B)\cap C=A\cap C+B\cap C$? I am looking for the most general conditions (if any) such that the equality stands. ...
0
votes
4answers
33 views

monotonic mapping from $(-\infty,\infty)$ to $(0,1)$

can someone please help me to find a mapping function which maps the whole real axis to (0,1). I want the function to be monotonic. Thanks in advance.
1
vote
1answer
21 views

If $S = \{x \in [0,1] \mid f(x) \neq0\}$ is a set, what is the complement of $S$, $S^{c}$?

If $S = \{x \in [0,1] \mid f(x) \neq0\}$ is a set, what is the complement of $S$, $S^{c}$? Is it: $S^{c} = \{x \neq [0,1] \mid f(x) \neq0\} = \{x \in \{\mathbb{R} \setminus [0,1]\} \mid f(x) \neq0\}$ ...
0
votes
0answers
17 views

The set of all polynomials of rational coefficient is countable [duplicate]

Prove that The set of all polynomials of rational coefficient is countable i don't know how help me
-1
votes
3answers
71 views

How many real valued Cauchy sequences are there? [closed]

Is the set of all Cauchy sequences of real numbers countable or uncountable? In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert ...
0
votes
3answers
30 views

If $A_1\cap…\cap A_n \neq \emptyset$, does $(A_1\cap…\cap A_n)^{c} =A_1^{c} \cup … \cup A_n^{c} = \emptyset$?

If I have some collection of sets such that $A_1\cap...\cap A_n \neq \emptyset$, then what happens if I apply the complement (denoted by superscript c) to both sides? i.e., $(A_1\cap...\cap A_n)^{c} ...
0
votes
2answers
28 views

What are $A_1 = \bigcup_{k \in \mathbb{N}} [0,k), A_2 = \bigcap_{k \in \mathbb{N}}(0,\frac{1}{k}),A_3 = \bigcap_{k \in \mathbb{N}}[0,\frac{1}{k})$

In the exercise it says "calculate" the following sets: $$A_1 = \bigcup_{k \in \mathbb{N}} [0,k)$$ $$A_2 = \bigcap_{k \in \mathbb{N}}(0,\frac{1}{k})$$$$A_3 = \bigcap_{k \in ...
0
votes
2answers
45 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 ...
3
votes
1answer
112 views

What is $\mathbb{R}^\mathbb{R}$

I do not know what it is. $\mathbb{R}$ is the set of real numbers. How come $\mathbb{R}\times\mathbb{R}\times \ldots $? Thanks.
1
vote
0answers
37 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
47 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
108 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
42 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
66 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
35 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
111 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
112 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
80 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
45 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
35 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
46 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 ...
0
votes
2answers
39 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 ...
1
vote
1answer
122 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
148 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
121 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 ...