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

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0answers
19 views
+50

Gelfand transform on disk algebra

I tried to prove that if $A$ is the disk algebra then the Gelfand transform is the identity map. The statement can be found here in Theorem 4.4 but it is given without proof. Please can someone check ...
13
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9answers
1k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
1
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1answer
27 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
1
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1answer
19 views

Proof by induction of a recursive sequence

I am studying CIE A levels Further Maths and I am stuck at a question from June 2002: Q The sequence of positive numbers $u_1,u_2,u_3,...$ is such that $u_1<4$ and ...
0
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2answers
39 views

If T(S) is linearly independent, show S is linearly independent

Let $T: V \to W$ be a linear transformation. Let $S = \{v_1,...,v_k\}$ and assume $T(S)$ is linearly independent. Show S is also linearly independent. I think I just have to prove that if $a_1 v_1 + ...
2
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2answers
32 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
2
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0answers
122 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
1
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1answer
29 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
1
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1answer
32 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
3
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0answers
46 views

$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt ...
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4answers
343 views

Is the reasoning/algebra for my proof correct? (musical tuning theory proof)

This isn't for a class, I was just wondering if I would be able to work out a proof for something like this myself for fun, and wanted to verify that my methods are correct. Basically, what I'm trying ...
0
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1answer
21 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
0
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0answers
23 views

A nowhere zero point in a linear mapping and Research Resources

Conjecture: If $\mathbb{F}$ is a finite field with at least 4 elements and $A$ is an invertible $n\times n$ matrix with entries in $\mathbb{F}$, then there are column vectors $x,y \in \mathbb{F^n}$ ...
1
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1answer
15 views

Dividing summations that have existing properties in each element

If $a_i/c_i > B$ for all $1 \le i \le k$, is it fair to assume that $(a_1 + a_2 + \cdots + a_k)/(c_1 + c_2 + \cdots + c_k) > B$ ? Is there a way to prove this? Thanks!
1
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0answers
33 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
1
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1answer
18 views

Proving a relation is a total order relation

Consider question #21 part a: Here is the solution: However, consider the definition of a total order relation: The solution didn't prove that the relation is a partial order relation. This ...
2
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2answers
21 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
2
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0answers
56 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
0
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0answers
19 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
1
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0answers
17 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
1
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2answers
52 views

Construct the truth table?

Any body help me .. How to solve this? (i) $(p\land q)\to (p \leftrightarrow (q \lor r))$ (ii) $(p \leftrightarrow q) \leftrightarrow ((p\land q) \lor (\neg q \land \neg p))$ (iii) ...
1
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1answer
17 views

Sylvester Gallai Problem

Recently I came through a book of Arthur Engel which mentioned a problem called Sylvester Problem which states that- A finite set $S$ of $n$ points in the Euclidean Plane has the property that any ...
0
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1answer
119 views

A group homomorphism proof with composition

Suppose I have groups $X$, $Y$, and $Z$, and I let $f: X \longrightarrow Y$ and $g: Y \longrightarrow Z$ be group homomorphisms. Now, I want to prove that $g \circ f : X \longrightarrow Z$ is a group ...
4
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1answer
35 views

Proof of $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x \to p}f(x) = l$

Let $f:(a,b) \to \mathbb{R}$ and $p \in (a,b)$. In proving the following implication, I am unsure about one step $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x ...
2
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0answers
20 views

Question about integration and sets of measure zero

Let $Q \subseteq \mathbb{R}^n$ be a box, $f: Q \to \mathbb{R}$ be bounded, integrable on $Q$. Suppose $g: Q \to \mathbb{R}$ is another bounded function such that $f(x) = g(x)$ for any $x \in Q ...
2
votes
1answer
61 views

Non-isomorphic unit groups

Show that the group $U_8$ of units modulo $8$ is not isomorphic to $U_{10}$? This is my answer. Check it for me please. Suppose $U_8$ is isomorphic to $U_{10}$, $3 ∈ U_{10}$ and order of $3$ in ...
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2answers
22 views

Proof of Conjugate Subgroup Isomorphism

Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that if $a$ is an element of $G$, then the subset $aHa^{-1} = \{g ∈ G | g = aha^-1 \text{ for some } h \in H\}$ is a subgroup of $G$ that is ...
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0answers
22 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
4
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1answer
85 views

The product of any three consecutive natural numbers is divisible by 9.prove or find a counterexample

can anyone give me feedback on my solution false counterexample: n(n+1)(n+2) let n= 1 1(2)(3) =6 is not divisible by 9 is this correct?
0
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0answers
36 views

Consecutive natural numbers [duplicate]

Please I want to know what is the most appropriate expression that if it is asked to find the counterexample of "The product of any three consecutive natural numbers is divisible by 9" My expression ...
3
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0answers
50 views

Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
0
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1answer
132 views

Abstract algebra, prove that $(a^m)^n$ =$ a^{mn}$

Let $a$ be an element of group $G$. For any integers $m,n \in \mathbb{Z}$ ($m,n$ can be positive and negative). Prove that $(a^{m})^{n}=a^{mn}$, then show that $(a^{-1})^{-1} = a$ by using what we ...
1
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1answer
25 views

Lang's proof of Cauchy's Theorem

In proving Cauchy's theorem in his 'Algebra', Lang first prove[s] by induction that if $G$ has exponent $n$ then the order of $G$ divides some power of $n$. Let $b \in G, b \ne 1$, and let $H$ be ...
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4answers
78 views

$f[A]\cap f[B]\supsetneq f[A\cap B]$ - Where does the string of equivalences fail ? [Chartrand 3E 9.12(b), 9.29]

I only realised that equality may fail in $f[A]\cap f[B]\supseteq f[A\cap B]$ (i.e., that we can have $A,B,f$ for which $f[A]\cap f[B]\neq f[A\cap B]$) after checking the answer. I don't see any ...
2
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1answer
16 views

Some questions about proof of Theorem 2.43 in Baby Rudin

I will include the proof here and highlight the parts that are giving me trouble. Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof ...
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1answer
40 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? Semsem kindly identified the problem: The normal to the disk is on the ...
3
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1answer
35 views

Where did this “+1” term come from for this inductive proof?

Where did this "+1" term come from for this inductive proof? It is in boxed in black. For context, We are trying to prove this sequence: has the following solution: $$x_{ n }=\frac { 3^{ n+1 ...
2
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0answers
66 views
+50

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
0
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1answer
71 views

Proof of The Associative Law.

The associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...
7
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1answer
53 views

The Limit: $\lim_{x \to \infty}\frac{e^{f(x+a)}}{e^{f(x)}}$

I'm doing some challenge review problems and I was wondering whether this proof looked correct: Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $\lim_{x \to ...
5
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3answers
156 views

Proving Undecidability of first order logic without first proving it for arithmetic.

All text I have read prove the Undecidability of first order logic a bit as an afterthought and after having proved the incompleteness and Undecidability of (Peano) Arithmetic. This proof also ...
0
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1answer
39 views

Find the Logical Inconsistency

Recently one of my friend came up with something which he claimed to be a proof of the famous Legendre Conjecture. Let me brief his argument. Statement of The Conjecture There exists at least ...
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0answers
23 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
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2answers
40 views

Show that $A^{(x,y)}$ is countable.

Question: Let $A$ be a countable set $A^{(x,y)}$ the set of all functions from $(x,y)$ to $A$. Show that $A^{(x,y)}$ is countable. My attempt: By proposition 7.1.2iii, $\mid B \mid^{\mid A \mid}$ ...
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0answers
21 views

A simple proof with directional derivatives

Suppose $\nabla f (x) \cdot d < 0$, prove that there exists $\delta > 0$ such that $$f(x + \tau d ) < f(x)$$ for all $\tau \in (0, \delta)$ My proof consists of only a few lines. ...
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0answers
33 views

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$. Show that $R$ is an equivalence relation.

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$ a. Show that $R$ is an equivalence relation. My attempt: By definition 6.2.3 $R$ is an equivalence ...
1
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0answers
33 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
2
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1answer
34 views

Locally nilpotent elements in modules

While reading through Lang's algebra, I came across following definition and proposition. In my opinion, something seems wrong. Some lemmas Lemma 1. Let $S$ be a multiplicative subset of $A$, and ...
4
votes
6answers
133 views

Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic

I'm reading a book about abstract algebra, but I'm having trouble solving this excercise: "Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic" Where $(\mathbb{Q}^*,\cdot)$ is ...
1
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1answer
25 views

Given two spanning trees of a graph, shows edges can be traded between them to create 2 new spanning trees

The precise problem: Let $T$, $T'$ be two spanning trees of a connected graph $G$. For $e \in E(T)\setminus E(T')$, prove that there is an edge $e' \in E(T') \setminus E(T')$ such that $T'+e-e'$ ...