1
vote
1answer
24 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
0
votes
2answers
25 views

Proving a set identity

If $A \subseteq X$ and $A_{\alpha}$ is a collection of all such subsets, I need to prove that: $$\left(\bigcap_{\text{all }\alpha} A_\alpha\right)^c = \bigcup_{\text{all }\alpha} A_{\alpha}^c$$ My ...
1
vote
3answers
41 views

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
1
vote
2answers
48 views

A nested cover of a set that eventually has infinite intersection with every infinite subset

Prove the following Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of sets with $$A_1 \subset A_2 \subset A_3 \dots $$ and $B \subset \bigcup_{n = 1}^{\infty} A_n$. If for every infinite subset $E$ ...
2
votes
3answers
64 views

Show that if $f$ is injective, then $f^{-1}(f(C))=C$ [duplicate]

Once again I am stuck on a question from Lay's Introduction to Analysis with Proof: Suppose that $f : A \rightarrow B$ and let C $\subseteq$ A and $D \subseteq B$. Show that if $f$ is injective, ...
1
vote
1answer
22 views

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions.

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions $f: X \rightarrow \mathbb R$ in each of the ...
2
votes
0answers
22 views

function over countable union, intersection, etc

is it true that for a give function, or a linear transformation, $L$, in $\mathbb{R}^n$, if $S=\bigcup_{j=1}^\infty T_j$, then $L(S)=L(\bigcup_{j=1}^\infty T_j)=\bigcup_{j=1}^\infty L(T_j)$? What if ...
0
votes
0answers
77 views

Proof of the cardinality of continuous functions from $[0,1]$ to $[0,1]$.

I've been thinking about the cardinality of continuous functions from $[0,1]$ to $[0,1]$. I know that the cardinality is the same as that of $[0,1]$ and the standard proof using the fact that such a ...
1
vote
0answers
19 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
1
vote
1answer
39 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
-2
votes
2answers
102 views

How do i construct $C^\infty$?

I'm trying to define $C^\infty$ rigorously and i have a trouble with this. Mathematical Induction should be used, but i dunno where to apply this. I'm going to illustrate what i tried below: Before I ...
5
votes
0answers
101 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
45 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
107 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
57 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
20 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
80 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
82 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
64 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
79 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
38 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 $L$ ...
3
votes
3answers
100 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
24 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
36 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
39 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
31 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
35 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
32 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
42 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
52 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
34 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
23 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: $S^{c} = \{x \neq [0,1] \mid f(x) \neq0\} = \{x \in \{\mathbb{R} \setminus [0,1]\} \mid f(x) \neq0\}$ ...
-1
votes
3answers
75 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
29 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
46 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
39 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
51 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 ...
-7
votes
1answer
112 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
73 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
36 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
114 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
118 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
81 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 ...