For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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1answer
43 views

Is the set $E=\{0.a_1a_2… \in \mathbb{R}\mid a_i= 4 \text{ or } a_i=7\}$ dense, compact or perfect?

I want to check my reasoning, I found that it's not dense but it's compact and perfect. $1$- It's not dense for 1 is neither in the set of a limit point of it. $2$- It's compact because it's both ...
0
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1answer
35 views

Is this proof with logarithmic exponentials correct?

I was unsure of this proof and some of the log rules I applied, could you check my proof and tell me if this proof is correct and if not, then what specifically is incorrect about the proof? ...
4
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2answers
56 views

How to prove the following bounds expression

Let n be a positive integer. Prove that there are 2^(n−1) ways to write n as a sum of positive integers, where the order of the sum matters. For example, there are 8 ways to write 4 as the sum of ...
0
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1answer
54 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
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2answers
39 views

Let $G$ be a group, $x∈G$, $ a,b∈\Bbb Z$ and $a⊥b$. If $x^a=x^b$, then $x=1$.

There is a missing step in this proof: http://math.stackexchange.com/a/106292/135812 Lemma Let $G$ be a group, $x\in G$, $a,b\in \mathbb Z$ and $a\perp b$. If $x^a = x^b$, then $x=1$. ...
0
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2answers
20 views

Choosing a committee with a constraint - where is my reasoning wrong?

Okay, this is an example from Challenge and Trill of Pre-college Mathematics by Krishnamurthy et al. In how many ways can we form a committee of three from a group of 10 men and 8 women, so that ...
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3answers
43 views

Show that if $A$ is any square matrix such that $A^n = 0$ for some positive intiger $n$, then $A$ is not invertible. (answer check)

Show that if $A$ is any square matrix such that $A^n = 0$ for some positive integer $n$, then $A$ is not invertible. I'm not sure if my proof is good enough, or enough "work" as my teacher put it ...
2
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0answers
54 views

Evaluating a limit…

I was solving a physics problem and this expression came about: $E =\lim_{N \to \infty} \left( \dfrac{k_0Q}{NR²}\displaystyle\sum_{i=0}^{(N/2-1)}\left[ \left( ...
0
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1answer
39 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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0answers
24 views

Set of semi infinite intervals on the real line is a topological basis

Show that the set $\{(r,\infty): r \in \mathbb{R}\}$ is a basis for a topology on the set of real numbers but not a topology itself. Any feedback on my proof would be appreciated. A collection of ...
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0answers
34 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...
2
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0answers
30 views

Find $f(n)$ in $\binom {2^n} {n^4} = (f(n)+ o(1))^n$

Task is to find $f(n)$ in the following equation: $\binom {2^n} {n^4} = (f(n)+ o(1))^n$ I've found that the problem is a bit over my head. I'm attaching my partial solution below: With use of the ...
1
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1answer
21 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
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0answers
43 views

Clarification: Rudin Theorem 3.7: Subsequential limits are closed

My question is this: Why does Rudin use $\delta$ in this proof? Would it not work just as well if $\forall i \ge1,$ $$x_{i}\in N_{2^{-(i+1)}}(q) \cap E^* $$ $$p_{n_i}\in N_{2^{-(i+1)}}(x_i) ...
0
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1answer
21 views

Solve $22x \equiv 5(mod 15)$

I looked at an example of this type, and here's my attempt: $gcd(22,15)=1$ and $1$ is a divisor of $5$ so solutions exist. Now $22x \equiv 5(mod 15)$ is the same as solving $22x=5$ in $Z_{15}$ i.e. ...
1
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2answers
69 views

Prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$

I have to prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$. My attempt: We consider the function $g(x)=x^{3}-3x+b$.Now since it is a polynomial it is ...
2
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1answer
34 views

If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$

If $p$ is prime and $\sigma(p^k) = n$, then $p\mid (n-1)$. proof: Suppose $\sigma(p^k) = [p^{k+1} -1]/(p-1) = n$. Then $n-1 = [p^{k+1} -1]/(p-1) - 1= [p^{k+1} -1 - (p-1)] /(p-1) = [p^{k+1} - ...
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0answers
34 views

Proving Replacement in $ZF^-$ without Replacement but with Collection

I was told in lecture, that in $ZF^-$ the replacement axiom scheme follows from adding the collection axiom scheme (without proof). So I tried proving it, but since I'm new to set theory, I need ...
0
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3answers
33 views

Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

My attempt: Let the characteristic be $n$. Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$, i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$ The least $n$ for which both are true is $30$, so $30$ ...
1
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1answer
30 views

abstract algebra one to one and onto

Let $G= \{(a,b) \mid a,b \in \Bbb Q \ , \ a \neq 0\}$ and a group under the operation $*$ defined by $(a,b)*(c,d)=(ac,ad+b)$ Suppose that $\varphi: G\to G$ defined by $\varphi((x,y))=(x^2,y)$. ...
2
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0answers
28 views

Norms on $\mathbb{R}$ seen as a $\mathbb{Q}$-vector space.

this is not really a question : I had some ideas on topics I don't feel secure with. I expose these hereafter : are there any mistakes in my reasonning ? Also, if anyone knows a good read about this ...
0
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1answer
19 views

How to prove if $G$ is a $k-connected$ graph that adding $y$ and its $k$ neighbors to $G$ is also $k-connected$

Thm. If $G$ is a $k-connected$ graph, and $G'$ is obtained by adding a new vertex $y$, with at least $k$ neighbors in $G$, then $G'$ is also $k-connected$. Here's what I have so far: There are 4 ...
1
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3answers
94 views

Proof verification: A $(16,5,8)$ binary code does exist.

Well I have used spheres in coding with radius, $r=\left\lfloor\frac{\delta -1}{2} \right\rfloor=\left\lfloor\frac{8 -1}{2} \right\rfloor = 3$ and that means we have $\sum \limits_{i=0}^3 {16 \choose ...
0
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2answers
26 views

Is there some trick to manipulating an equation? (adding 0s, multiplying by 1, etc..)

I have such a hard time doing this sort of thing that it's annoying me. I'm not very mathematically inclined but it frustrates me that a solution with such a small answer takes me more than a page to ...
0
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0answers
30 views

What's the correct logical conclusion after proving a value holds?

I had to prove that for every set $s$, the number of subsets with odd cardinalities is $2^{n-1}$. I concluded that this formula holds everytime $|s| \geq 1$ and then I used an inductive process to ...
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0answers
23 views

Is this a valid proof of surjectivity?

Problem: Determine whether the function $f: [3, \infty) \to [5,\infty)$ given by $f(x) = (x-3)^2+5$ is surjective. Answer. It is surjective. Proof: Let $\omega \in [5,\infty)$. I claim that ...
1
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0answers
28 views

BusyBeaver growth: “simple” proof

I just try to prove that $BB(n)$ (BusyBeaver-Function) grows faster than any other computable function. Maybe someone can check the proof? $f(n)$ is a computable function which grows to infinity: ...
1
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3answers
74 views

Proof that a sequence is convergent

I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I proved that it is increasing through ...
3
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1answer
40 views

Prove $\forall n \in N$, every set of natural numbers of size n has a maximum element. May assume that sets do not repeat numbers.

Prove using induction. So i'm a bit confused about how to do this question. My attempt at it seems like i'm missing a lot and it looked to easy. ...
1
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1answer
26 views

For integer $n$ prove that if there is no integer $m\le \sqrt{n}$ such that $ m | n$, then $n$ is prime.

At first I thought that the best way to prove this statement is to take the direct approach and show the subset {1, 2, 3,...sqrt(n)} and the subset {sqrt(n),... n/3, ..., n/2,...,n} and show that ...
1
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1answer
14 views

Order-preserving function

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x^2)+1$. I have ...
2
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1answer
18 views

Proof Verification : If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ closed in $S$ .

If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ open in $S$? Attempt: $U(x)$ is the union of all connected subsets of $S$ containing $\{x\}$. Hence, $U(x)$ ...
2
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1answer
224 views

True or false: Surjective linear transformation proof verification

Is it true that if $f:V \to W $ is a surjective linear transformation, then the cardinality $|f^{-1} (w_1)| = |f^{-1}(w_2)|$ for all $w_i \in W$? Here $f^{-1}(w) := ${$v \in V | f(v) = w$}. If it is, ...
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1answer
17 views

Is Z2 is simple.

Is Z2 simple.Does every subgroup of Z2 simple. Solution. Z2 is simple, however subgroup of order 1 in Z2 is not simple.
0
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2answers
18 views

GCD(m,n) = sm + tn proof

Suppose that m and n are positive integers and that s and t are integers such that gcd(m,n) = sm + tn. Show that s and t cannot both be positive or both be negative. I understand that if both of them ...
0
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3answers
35 views

$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$

Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say ...
1
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1answer
44 views

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I've come up with a proof for the following question, and I'd like to know if it's correct (as I couldn't find anything online along the lines of what I did). Question Let $p$ and $q$ be primes ...
2
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3answers
63 views

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
1
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0answers
55 views

Proof of Definite Integral of Even Function for Improper Integrals

I am trying to prove $\displaystyle \int_{\mathop \to -a}^{\mathop \to a} f \left({x}\right) \ \mathrm d x = 2 \int_0^{\mathop \to a} f \left({x}\right) \ \mathrm d x$ for $f$ which is an even ...
3
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1answer
135 views

Did I construct an infinite set equal to $\{1\}$?

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU He tries to explain why he things infinite sets don't make ...
0
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1answer
20 views

Continuity and Subspace Topology

I think the first one is false. If we let $(-1/2, 1/2) \subset \Bbb R$ and $(0,1/4) \subset \Bbb R$, then for $f(x) = x$ defined on $[0,1) \subset M = \Bbb R$, we have $f^{-1}(-1/2, 1/2) = ...
0
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1answer
40 views

Proofs in Stochastic Processes

Let $$X_{n}$$ be an irreducible Markov chain on the state space {1,...,N}. Show that there exists $$C < \infty$$ and $$\rho < 1$$ such that for any states i,j, $$\mathbb{P} [ X_{m}\neq j , m=0 ...
0
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1answer
38 views

Let $\alpha$ and $\beta$ be disjoint cycles. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$

Let $\alpha$ and $\beta$ be disjoint cycles. Say $\alpha = (a_1a_2...a_s)$, $\beta=(b_1b_2...b_r)$. Prove for every positive integer n, $(\alpha\beta)^n=\alpha^n\beta^n$ My proof is as follows: ...
0
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0answers
20 views

Find the derivative of $f(z)=\frac{1}{z^3}$, where $z \ne 0$, in two different ways

Find the derivative of $f(z)=\frac{1}{z^3}$, where $z \ne 0$. Find the derivative using the definition of derivative Find the derivative using Cauchy Riemann equations Using the definition of ...
2
votes
1answer
38 views

continuity of $\max \{f_1(x),f_2(x),\cdots,f_m(x) \}$ at $a$

let $f_1,\cdots, f_m$ be a real valued functions defined on a set $S$ in $\mathbb R^n$. Assume that each $f_k$ is continuous at the point $a$ of $S$. For each $x \in S : f(x) = \max ...
0
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0answers
17 views

Trouble with minor detail in proof

We proved in class the countable subadditivity of a general measure. My question is at the end. Statement: If $\{A_k\}_{k=1}^{\infty} \subseteq \mathscr{F}$ and $\cup_{k=1}^{\infty}A_k \in ...
1
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0answers
50 views

Big-O estimate (smallest order)

I'm trying to give a big-O estimate for each of these functions, where I want to use a simple function $g$ of smallest order. I have them all done I just wanted to someone to run through and check ...
2
votes
1answer
4k views

Probabalistic proof of green-eyed dragons logic puzzle

I came across the "green-eyed dragons" puzzle (alternatively known as the "blue eyed villagers" puzzle). The typical proof uses a straightforward inductive strategy. I came up with a probabalistic ...
1
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2answers
72 views

Show that the function $f$ is continuous only at the irrational points

Prove that the function $f$ is continuous only at the irrational points. $f(x)=\begin{cases} 0 & ;x \in \mathbb R-\mathbb Q \\ \dfrac{1}{n} & ;x=\dfrac {m}{n} :\gcd(m,n)=1;m,n ...
1
vote
2answers
16 views

For magic squares prove $id_v+r^2=2c $

I've been presented with the following problem: Let $V$ be the vector space of all $3\times3$ magic squares. Let $r:V\rightarrow V$ be the linear image which rotates a magic square $90^\circ$. Let ...