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

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2answers
29 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
2
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1answer
24 views

Uniform Continuity implies Continuity

Let $f$ be a function from a metric space $X$ to a metric space $Y$. Show that if $f$ is uniformly continuous on $X$ then $f$ is continuous on $X$. Show that the converse is not true. Uniform ...
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2answers
28 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
0
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0answers
35 views

Help fix this proof.

What is wrong with this proof? I followed the example of the answer to another one of my questions, here Define a general recurrence relation as $$f(x)^2=A(x)+B(x)f(x+n).$$ Substitute the root ...
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1answer
25 views

My problem in the definition of Dirichlet generating function?

In the definition of Dirichlet generating function "for the square-free numbers " is: $$ \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty} \frac {|\mu(n)|}{n^s} $$ where $\mu$ is Moebius ...
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1answer
28 views

If $A\subseteq\Bbb R$ is nonempty with $|A|\ge 2$, then $A$ totally disconnected $\iff A^\circ=\emptyset$.

In the course of working on an exercise, I came up with the claim given in the title. Just looking for verification. $\underline{\text{Claim: } A\text{ is totally disconnected}\iff ...
2
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1answer
31 views

Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...
3
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3answers
78 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
1
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1answer
23 views

Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
1
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0answers
22 views

Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of ...
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1answer
75 views

Injections, Surjections, Bijections [on hold]

So i was given a question that asks me to determine whether the function is injective, bijective, or surjective. If you answer bijective than determine the functions inverse, domain, and target space. ...
2
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2answers
47 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
4
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1answer
94 views

Possible proof of Fermat's Last Theorem for prime exponents greater than 2

I would appreciate if someone could check my attempt in proving the Fermat's Last Theorem for prime exponents greater than $2$. Firstly, let's prove a couple of lemmas which state that sum or ...
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2answers
31 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
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2answers
22 views

Spot the error in experimenting with contradiction on 5's rationality.

Let $5=\frac ab$ $\forall\ a,b\ \epsilon\ N$. And $(a,b)=1$ Squaring both sides, $25b^2=a^2$ Thus, $25|a^2$; $25|a$ So $a=25m$ Substituting, $25b^2=25^2m^2$ So $b^2=25m^2$ So $25|b$ (By the ...
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0answers
66 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective [on hold]

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
0
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1answer
25 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
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0answers
43 views

prove that for any 2n≥2 and any \a ​1 ​​ ,…,a ​n ​​ ∈N, we have the following: [on hold]

So the question I was given goes like this we will introduce a mystery function,P:N→N. We don't know a formula for P (and we won't be able to determine one!) but we do know that P satisfies the ...
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3answers
70 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
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3answers
73 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
0
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3answers
58 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
2
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2answers
24 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
0
votes
3answers
34 views

Trigonometical identity proof

I was given a proving sum: $\sec(x) + \tan (x) = p$, prove $\frac{p^2-1}{p^2+1} = \sin (x)$ I went head on and tried to directly do it by solving the LHS: $\sec(x) + \tan(x)$ = $\frac{1}{\cos(x)} ...
1
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1answer
28 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
3
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4answers
97 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
0
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0answers
31 views

Proof Check In: Prove that $(\mathbb{Z}_n, +)$, the integers (mod $n$) under addition, is a group.

I received some help and direction on this from some users a few days ago, and have tried to take that information and craft it into something proofy. I would appreciate general suggestions, edits, ...
2
votes
3answers
48 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
0
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1answer
19 views

Composition of analytic function with arithmetic function

Consider an arithmetic function $g$ with codomain $\{a,b\}$ and a function $f$ which is analytic on some domain including $\{a,b\}$. We therefore have $$f(g(n))=\sum_{k=0}^\infty c_k (g(n)-a)^k$$ and ...
2
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1answer
17 views

Assumption on characteristics in an exercise about roots of unity

I'm solving the following exercise: "Let $K$ be a field, $char(K) \nmid 2n$ for $n \geq 1$ an odd integer. If $K$ contains a primitive $n$-th root of unity, then it also contains a primitive $2n$-th ...
2
votes
5answers
76 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
0
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1answer
52 views

Prove that $(a,b]\subseteq \mathbb{R}$ is not open.

I want to prove myself that a half-interval $(a,b]\subseteq \mathbb{R}$ is not an open set. I checked it in here. My proof: We wish to prove that $b\notin (a,b]^{\circ}$. Assume that $b\in ...
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0answers
13 views

Extending orthogonal representation of $SU_2$ to $U_2$

Let $\phi: SU_2 \rightarrow SO_3(\mathbb{R})$ be the orthogonal representation of $SU_2$, obtained by letting $SU_2$ act on the three-dimensional vector space of trace-zero skew-hermitian matrices. ...
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2answers
14 views

A finite set and the set of its fixed points under any involution have cardinalities of the same parity

I am trying to write down a formal proof of the following fact: Let $A$ be a non-empty finite set and $f$ an involution on $A$. If $A'$ is the set of fixed points of the involution $f$, then $|A| ...
2
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3answers
61 views

Prove that $\sum_{d|n}\phi(d)=n$ where $\phi$ is the Euler's phi function, $n,c\in\mathbb{N}$

Here is a very elementary number theory proof using strong induction. Please mark/grade. Prove that $$\sum_{d|n}\phi(d)=n$$where $\phi$ is the Euler's phi function, $n,d\in\mathbb{N}$ First, ...
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2answers
44 views

Proving that any connected graph has a vertex whose removal results in a connected graph

I want to prove that: for any simple, connected graph there is at least one node whose removal results in a connected graph. Here is my proof: Suppose that a graph $G$ is simple connected graph with ...
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2answers
64 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
3
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1answer
19 views

Question about assumptions for Picard-Lindelof Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
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1answer
23 views

Asymptote criterion

Let $f:(a, \infty)\to \Bbb R$ be a differentiable function such that exists $\lim_{x\to\infty}f(x)=l<\infty$ and exists (in the sense it can also be infinity) $\lim_{x\to\infty}f'(x)$. Under these ...
3
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1answer
27 views

The set is closed (resp. open) iff the complement set is open (resp. closed)

There's a theorem in my small danish course book. Let $(M,d)$ be a metric space. Theorem: The concepts of open and closed are dual: A set $A\subseteq M$ is closed (resp. open) if and only if the ...
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1answer
29 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
0
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2answers
58 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
2
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1answer
22 views

Number of square matrices of order $n$ where each row and each column has at most one $1$

What is the number of square matrices of order $n$ with the property that each row and each column has at most one $1$, and $0$s elsewhere? For example, when $n=2$, there are $7$ such matrices: ...
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1answer
25 views

Proving B Congruent C given AB congruent AC

This is a very trivial question, i seem to have arrived at a proof for an excercise but the proof just doesn't feel.. right. It is too small and simple. The fact to be proved is that if $AB\equiv AC$ ...
0
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0answers
32 views

Separated subsets of $\mathbb{R}^k$

Let $A$ and $B$ be separated subsets of some $\mathbb{R}^n$, suppose $a\in A, b\in B,$ and define $p(t)=(1-t)a+tb$ for $t\in \mathbb{R}^1$. Put $A_0=p^{-1}(A), B_0=p^{-1}(B)$. (a) Prove that $A_0$ ...
0
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0answers
18 views

Isomorphic Fields Leading to Isomorphic Splitting Fields

Link to Original Text: Theorem 10.6 Let $F, F'$ be two fields isomorphic via $\varphi$. Suppose that $f = \sum_{i=0}^m c_ix^i \in F[x]$ splits in $E$, and that the corresponding $f' = \sum_{i=0}^m ...
2
votes
1answer
49 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
1
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1answer
45 views

A proof that every set of natural numbers contains a minimal element

I'm currently trying to extend my basic knowledge and in order to do so, I started with the Peano-axioms. I think, I understand the underlying thoughts and I want to prove the following theorem using ...
1
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1answer
63 views

What is wrong with my brute-force approach to proving that $\mathbb R$ as a metric space obeys the triangle inequality?

In a self-study of metric spaces, I'm looking at the very basic exercise of proving that $(\mathbb R, |y-x|)$ is a metric space. The sticking point was the triangle inequality. I did manage to ...
2
votes
1answer
58 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
0
votes
2answers
74 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...