Tagged Questions
1
vote
2answers
56 views
Proving Bijections in $\mathbb{R}^n$
I have a question that may seem trivial or silly, however I'll try to make my point clear. I'm a student of Mathematical Physics and unfortunately my college doesn't offer a set theory course, so that ...
0
votes
1answer
39 views
Showing that a union of the subsets of two $\sigma$-algebras is a $\sigma$-algebra.
I got back an assignment for a first course in analysis and I have made a very basic error, and I'm having a lot of trouble pinpointing exactly what piece of information I'm missing.
You have two ...
5
votes
2answers
92 views
Showing ${\frak c}:=|\Bbb{R}|=2^{\aleph_0}$ using only the axioms of the reals
It is a well-known fact that the cardinality of the continuum, ${\frak c}:=|\Bbb{R}|$, is the same as the cardinality of the powerset of the natural numbers, but I've tasked myself with proving the ...
1
vote
2answers
70 views
Proof by counter example to false analysis statements.
(i)Disprove that the isolated points of a set form a closed set.
(ii)Disprove
every open set contains at least 2 points.
(iii) Disprove $\partial ( S \cup D) = \partial S \cup \partial D$
$S= ...
0
votes
0answers
70 views
Show that the fiber $f^{-1}(a)$ is finite if $a∈ℝ,a≠0$
Let $f:ℝ→ℝ$ be a real analytic function. If $f$ has infinitely many zeros, then we know that the fiber $f^{-1}(0)$ is an infinite discrete and countable set. Let $a∈ℝ,a≠0$, we know also that the fiber ...
0
votes
3answers
86 views
How can I prove that a set of real numbers always have a minimum?
If I take a set of real numbers, say $S$, can I always prove that there is a number $n^*$, contained in the set $S$, which for all other $n$ in that same set, $n^*\leq n$ always holds?
I guess this ...
0
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0answers
58 views
Simple proof of well ordering principle
I've seen some proof of the well-ordering principle and they all use the contradiction method.
However, direct proof would be simpler.
Can you check for my proof?
I will use mathematical ...
1
vote
3answers
88 views
Empty intersection and empty union
If $A_\alpha$ are subsets of a set $S$ then
$\bigcup_{\alpha \in I}A_\alpha$ = "all $x \in S$ so that $x$ is in at least one $A_\alpha$"
$\bigcap_{\alpha \in I} A_\alpha$ = "all $x \in S$ so that ...
4
votes
2answers
55 views
Intuition behind the difference between derived sets and closed sets?
I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
0
votes
1answer
55 views
Countability of $E_1 \times E_2 \times …$
True or False: If $E_1, E_2,...$ are finite sets and $$E:= E_1 \times E_2 \times ...\,\, := \left\{(x_1,x_2...): x_j \in E_j\,\,\forall j \in \mathbb{N}\right\}$$ then $E$ is countable.
Attempt: ...
1
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3answers
31 views
A question about countability of a set
Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of ...
5
votes
2answers
59 views
The cardinality of Lebesgue sets
Suppose $A=\{S\;|\;S \subset \mathbb R^n, S\text{ is Lebesgue measurable}\}$. What is the cardinality of $A$? Is it the same as the cardinality of all of the real numbers?
3
votes
7answers
209 views
Doubt on rational and real numbers
I am going through the numbers system from an analysis book. It is written that:
1) there is no rational number $p \ ( > 0)$ which satisfies $p^2=2$.
2) The set $\{p: p^2 < 2\}$ does not have ...
1
vote
1answer
47 views
About the existence of bijections between $(0,1)^{r}$ and $ℝ^{r}$
I know that there is some bijections between the open unit interval $(0,1)$ and $ℝ$. My question is about the existence of bijections between $(0,1)^{r}$ and $ℝ^{r}$ for $r∈ℕ$.
1
vote
1answer
34 views
A proof on infinium property
Suppose that $A_1$ and $A_2$ are nonempty sets with $A_2 \subset A_1$ of real numbers that are bounded below. I want to show that $\inf A_1 \leq \inf A_2$
I managed to do the opposite, that is $\inf ...
3
votes
2answers
49 views
How do I simplify the following union of sets?
Am a bit confused on how a union of sets can be arrived at, am not sure how to start even and some guidance is much appreciated, below is my attempt
$$\bigcup_{n\in\mathbb{N}} ...
1
vote
6answers
99 views
Proving that an injective function is bijective
I am having a lot of trouble starting this proof. I would greatly appreciate any help I can get here. Thanks.
Let $n\in \mathbb{N}$. Prove that any injective function from $\{1,2,\ldots,n\}$ to ...
4
votes
3answers
112 views
An odd proof of the uncountability of the reals
So, in proving that the reals are uncountable you assume otherwise and attempt to make a list of them. Working in binary this list is of form $a_1,a_2,\ldots$ where $a_i = x_{i,1}x_{i,2},\ldots$ ...
5
votes
3answers
75 views
How to derive a union of sets as a disjoint union?
$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$
The results is obvious enough, but how to prove this
0
votes
1answer
37 views
Question about equivalence relation
$A_n=\{1,2,3,\dots,n\}$ and $B$ is a nonempty set, define $$\prod_{a\in A_1}B:=B\;,$$ for $n>1$ we have $$\prod_{a\in A_n}B:=B\times\prod_{a\in A_{n-1}}B\;.$$ What exactly is $\prod_{a\in A_3}$ ...
0
votes
2answers
81 views
Find all properties of this set $S$ in the metric space $(\mathbb{R},\rho)$
Set $S=(0,1]\cup([-\sqrt{2},\sqrt{3})\cap\mathbb{Q})$ in metric space $(\mathbb{R},\rho)$ with standard metric $\rho(x,y)=|x-y|$.
Find the following sets:
$S'$, $\text{Closure}(S)$, ...
2
votes
1answer
104 views
Bounded metric space $(X,\rho)$ question for any $S\subset{X}$?
Prove that in any metric space $(X,\rho)$ for any $S\subset{X}$, we have $bd(bd(S)) = bd(bd(bd(S)))$, while not necessarily $bd(S)=bd(bd(S))$
Prof's Hint (first show that a boundary of a closed set ...
2
votes
4answers
114 views
Is there a good visual aid or picture to help understand openness and closedness?
I'm struggling to grasp the idea of open, closed, clopen and not open and not closed sets in a more formal approach like how it's described in a math analysis class. Is there a good picture somewhere ...
3
votes
1answer
111 views
Proof $\{x \in \mathbb{R} \mid 0 \leq x \leq 1\}$ in uncountable
Prove that the interval $\{x \in \mathbb{R} \mid 0 \leq x \leq 1\}$ in uncountable. In other words show that no function $f\colon \mathbb{N} \to [0,1]$ can have one to one and onto correspondence.
...
0
votes
2answers
105 views
relationship between Zorn's lemma and Axiom of Completeness
For me , they look like they are 'similar' to each other , just that one is used in set and another one is used in numbers. Can anyone tell me is there any relationship between Zorn's Lemma and Axiom ...
0
votes
1answer
82 views
Exterior, Interior, Boundary
If we denote the general point of $\mathbb{R}^2$ by $(x,y)$, determine $\operatorname{Int}A$, $\operatorname{Ext}A$, and $\operatorname{Bd}A$ for the subset $A$ of $\mathbb{R}^2$ specified by each ...
1
vote
2answers
55 views
Why is $\bigcup_{n=1}^{\infty} [a+\frac{1}{n},b-\frac{1}{n} ]=(a,b)$?
Why is $\displaystyle \bigcup_{n=1}^{\infty} \left[a+\frac{1}{n},b-\frac{1}{n} \right]=(a,b)$? Isn't it $\displaystyle \lim_{n \to \infty} \frac{1}{n} = 0$, which implies the union is $[a,b]$?
I am ...
6
votes
1answer
221 views
Bolzano-Weierstrass and measures
Let $\{\mathcal{A}_n\}$ be an infinite sequence of sets with $\mathcal{A}_n \subset \mathcal{M}$, where $\mathcal{M}$ is a bounded subset of $\mathbb{R}$ (for simplicity). Is there a "nice" limit ...
2
votes
1answer
197 views
Perfect Set and Compact Set
I am having some difficulty in understanding the difference between Perfect and Compact sets. More specifically, my problem is rather understanding how Perfect sets are different from Compact sets, by ...
1
vote
4answers
233 views
The use of ‘infinity’ in a intersection/union of infinitely many sets.
I understand the multiple meanings of infinity, per example, the difference between $\aleph_0$ and the $\infty$ in calculus limits as explained here: The Aleph numbers and infinity in calculus.
...
0
votes
1answer
27 views
Showing equivalence with collection of subsets…
Let $f:X\rightarrow Y$ where $X$ and $Y$ are sets. Prove that if {$S_\alpha$}$_{\alpha\in I}$ is a collection of subsets of $Y$, then $f^{-1}(\cup_{\alpha\in I}S_\alpha)=\cup_{\alpha\in ...
1
vote
1answer
95 views
The complement of Cantor set over closed interval 0 to1. What is its measure and closure??
Is the complement of Cantor set $C$ still measure zero? Meanwhile, I know its accumulation point is $C$ itself (right?). So its closure would be $C$, correct? Why??? Notice: I am asking for the ...
6
votes
2answers
94 views
Proof that$ (a+A)\cap A=\varnothing$
I finished my test and there is a question I completely failed but that my teacher did not go over, so I was hoping someone could post a correction of it, so that I understand what I was supposed to ...
1
vote
2answers
96 views
set of all points where $|f(x)|>\epsilon$ is finite
$f:\mathbb{R}\rightarrow \mathbb{R}$ is function such that $\forall\epsilon>0$, the set $\{x:|f(x)|>\epsilon\}$ is finite. We need to show $\{x:f(x)=0\}$ is uncountable. Could any one give me ...
0
votes
1answer
52 views
Summation of Supremum
Let $[a,b]$ be an interval in $\mathbb{R}$.
Let $P=\{x_0,...,x_n\}$ be a partition of $[a,b]$ and $f$ be a real function bounded on $[a,b]$.
Let $\alpha$ be a monotonically increasing function on ...
4
votes
2answers
356 views
Bijection from $\mathbb R$ to $\mathbb {R^N}$
How does one create an explicit bijection from the reals to the set of all sequences of reals? I know how to make a bijection from $\mathbb R$ to $\mathbb {R \times R}$.
I have an idea but I am not ...
1
vote
3answers
154 views
proof: set countable iff there is a bijection
In class we had the following definiton of a countable set:
A set $M$ is countable if there is a bijection between $\mathbb N$ and $M$.
In our exam today, we had the following thesis given:If $A$ is ...
0
votes
3answers
133 views
Supremum and Infimum
Is this sufficient? Also, any good books/other suggestions regarding the subject will be very helpful.
Find min, max, inf, sup (if they exist):
$$B=\left\{\frac{m}{m+n}:m,n\in\mathbb{N}\right\}$$
...
0
votes
1answer
126 views
Every non-empty subset of $\mathbb{R}$ bounded above has a largest element
I restarted my analysis book from page 1 trying to relearn everything because I feel like my knowledge is too fragmented. This true false question asks exactly what the title says. I don't know 100% ...
11
votes
1answer
245 views
Constructing a bijection from (0,1) to the irrationals in (0,1).
How does one construct a bijection from (0,1) to the irrationals in (0,1)? Or if I am getting my notation right, can you provide an explicit function $f:(0,1)\rightarrow(0,1)\backslash\mathbb{Q}$ such ...
5
votes
2answers
73 views
Countable $A \subseteq \mathbb{R}$ satisfy $(x+A) \cap A = \emptyset$
I have to prove that
if $A \subseteq \mathbb{R}$ is countable, then
$\exists x \in \mathbb{R}. (x+A) \cap A = \emptyset $
where $(x+A)$ denotes the set $\{x + a |\: a \in A\}$.
I can see ...
11
votes
3answers
317 views
Cantor set and countability.
The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably ...
14
votes
5answers
892 views
Importance of Axiom of Choice
First a quick question regarding the definition of the axiom of choice. Do the sets have to be mutually disjoint nonempty sets or just non-empty? One source states: "For any set X of nonempty sets, ...
0
votes
2answers
266 views
Closed under countable union
I am reading a tutorial on measure theory and it states: "Given an interval $E = [a, b]$ and a set $S$ of subsets of $E$ which is closed under countable unions, we define the following..."
I was ...
2
votes
4answers
218 views
Infinite set and countable subsets
Prove that a set $A$ is infinite if and only if $A$ contains a countable subset $C$.
I know I have to build a sequence and then I'll get a countable subset, but I don't know how to build that sequence ...
4
votes
5answers
657 views
Countable subset of a uncountable set
Is it true that for any uncountable subset T of $\mathbb R$, one can find a subset S of T such that S is countable. If yes, how can we prove it?
Thanks!
Edit: Is there a countable subset S of T such ...
16
votes
9answers
2k views
Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
3
votes
2answers
116 views
(Help with) A simple yet specific argument to prove Q is countable
I was asked to prove that $\mathbb{Q}$ is countable. Though there are several proofs to this I want to prove it through a specific argument.
Let $\mathbb{Q} = \{x|n.x+m=0; n,m\in\mathbb{Z}\}$
I ...
6
votes
1answer
182 views
On cardinality of equivalence classes
Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they ...
1
vote
1answer
191 views
How to prove a set is a Dedekind Cut?
In my Real Analysis class we just started talking about Dedekind cuts and I'm very confused on how to prove this one homework assignment: Given the set $-A=\{-(a+b):a\in\mathbb{Q}^+, ...
