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Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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42 views

Chebyshev Polynomials: Properties of Derivatives

Show that: $T_n'(x)$=$2n\sum_{k=0\\k+n~~odd}^{n-1}\frac{1}{c_k}T_k(x)$ $T_n''(x)$=$\sum_{k=0\\k+n even}^{n-2}\frac{1}{c_k}n(n^2-k^2)T_k(x)$ where $c_0=2$ and $c_n=1$ for $n\geq1$ I tried using the ...
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0answers
37 views

Elementary Proof on Perfect Squares.

The Proof I'm working on is: If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational.\ The farthest I've gotten is by proving by contradiction, assuming that $\sqrt{r}$ is ...
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0answers
30 views

Differentiability and continuity while partials have different conditions

The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question). Also, let me add two other links ...
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0answers
48 views

Number of points on the elliptic curve $\ y^2 = x^3 + 1$ [duplicate]

Consider the elliptic curve defined by $\ y^2 = x^3 + 1\ $ over $\ \mathbb{Z}_p,\ $ where $\ p \equiv 2 \pmod{3}\ $ is prime. Prove that the number of points on the curve is exactly $\ p + 1.\ $ Hint:...
1
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1answer
37 views

linear algebra eigenvalue proof question

Prove that a 2 × 2 matrix with just one eigenvalue (of multiplicity two) is diagonalizable if and only if it is already a diagonal matrix. So, by multiplicity two, I'm guessing it just means you have ...
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0answers
21 views

A question about space quotient and homeomorphism [duplicate]

Let $X$ and $Y$ be topological space and let be $\sim_X$ and $\sim_Y$ the corresponding equivalence relations on $X$ and $Y$. Let $f\colon X\to Y$ an homeomorphism, suppose we are in the following ...
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3answers
69 views

Proof by contradiction, status of initial assumption after the proof is complete. [duplicate]

First of all I'd like to say that I have looked for the answers to my specific question and have not found it in the existing topics. The question is fairly simple. Say, we need to prove statement P ...
3
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3answers
74 views

Combinatorics Proof of $\sum_{i=0}^n \sum_{j=0}^{i-1} j = {n+1 \choose 3}$

Proof of $\sum_{i=0}^n \sum_{j=0}^{i-1} j = {n+1 \choose 3}$ I am trying to generate a combinatorics proof of this identity, but have been stuck for hours. I've been trying to think of someway to ...
3
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0answers
33 views

Prove that matrix $A = \sum_k^{n}x_k\lambda_ky_k^T$

Prove that matrix $\ A = \sum_{k=1}^n {\bf x}_k \lambda_k {\bf y}_k^T.\ $ If $\ {\bf x}_k\ $ and $\ {\bf y}_k\ $ are the corresponding eigenvector from the left and from the right of $\lambda_k$. Any ...
2
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3answers
71 views

Finding $\lim_{x \to \infty} \sqrt{x} c^x$ for $0<c<1$ [duplicate]

Is there a short way to prove that $\lim_{x \to \infty} \sqrt{x} c^x = 0$ for $0<c<1$? I tried using L'Hospital's rule and a few substitutions, and even if I was getting somewhere the proof was ...
1
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1answer
36 views

Showing that the simple continued fraction of $\sqrt{d}$ has period length 1 iff $d=a^2+1$

Given that I know if $d$ is an integer that $\sqrt{d}=[\alpha_0,\bar{\alpha_1},...\bar{\alpha_n},\bar{2\alpha_0}]$. I want to show that $\sqrt{d}$ has period length 1 if and only if $d=a^2+1$, for ...
1
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1answer
49 views

The units digit of a perfect square is 6. What are the possible values of the tens digit? [closed]

I know the answer to this already: the possible values of the tens digit are 1, 3, 5, 7, and 9. But I don't know how to prove it, can someone help please? Thanks!
2
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5answers
81 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
7
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1answer
85 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+...
2
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1answer
48 views

Attacking proof of a statement with the form: $(\forall x \in X): ( P_1(x) \lor P_2(x) ) \rightarrow Q$

I have a statement of the form $(\forall x \in X): ( P_1(x) \lor P_2(x) ) \rightarrow Q$ but I am not sure how to approach proving it. I feel as though there is some sort of case analysis that can be ...
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1answer
34 views

Limit of $\sqrt[n]{n^k}$ for k $\in \mathbb{N}$ for $n \rightarrow \infty$

So the title basically says what my question is I'm looking for a proof for the limit of the given sequence (title).
2
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1answer
42 views

How to prove this simple property for two sets?

We are given two vectors, $a = (a_1,\dots,a_n)$ and $b = (b_1,\dots,b_n)$ such that $0 \le a_i=b_i \le \varDelta$ for $i=1,\dots,n$. We want to modify each of these vectors in an iterative procedure ...
0
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3answers
57 views

Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$

Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties $u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
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0answers
26 views

Proving the uniqueness of x=sqrt(r)

Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$ I would appreciate any nudge in the ...
0
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1answer
29 views

How to cite a theorem in the proof of another theorem?

I want to prove a theorem using the result of a well-known theorem. Should I write the well-known theorem as a lemma since it aids the proof of my theorem or as a theorem?
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0answers
10 views

Proper way to write a theorem with approximation result.

I have a lemma that shows that we can approximate $a$ with $b$ if $n \gg 1$ and I use the result to prove a theorem. Which one of the following ways is better to write this lemma? Provided that $n \...
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0answers
33 views

Help with elementary proof about r in the real numbers

If $r<0$, there exists no $x \in \mathbb{R}$ such that $x^2 = r$. I'm thinking I need to prove by contradiction assuming there does exist an $x$ such that $x^2=r$, but I'm having trouble finding ...
0
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0answers
37 views

A question regarding the Galois extension of the cyclotomic polynomial $\Phi_{15}$

Given the cyclotomic polynomial $\Phi_{15}$, I am trying to : i) Determine the isomorphism type of the Galois group of $\Phi_{15}$ over $\Bbb Q$. ii)Letting ω be a primitive 15-th root of unity in $\...
2
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3answers
40 views

Proving $N_2$ is not normal subgroup of $H_2$ if $\phi$ is not surjective

I am given that $\phi: H_1 \to H_2$ is a non-surjective group homomorphism and $\phi(N_1) = N_2$ where $N_1 \unlhd H_1$. How do I prove that $N_2$ may not be a normal subgroup of $H_2$? Attempt: ...
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1answer
19 views

How do I use induction to prove a claim of a recursive set definition?

The set X is defined as 12 ∈ X 15 ∈ X if x, y ∈ X, then x + y ∈ X if x, y ∈ X, then x − y ∈ X Claim: for every natural number n, 3n ∈ X I know I should induct on natural numbers that means my base ...
4
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1answer
48 views

$\cos(n\vartheta)=\frac{a_n}{3^n}$

I want to show, if i know that $\cos(\vartheta)=\frac{1}{3}$ than $\cos(n\vartheta)=\frac{a_n}{3^n}$ for $n\in \mathbb{N}$, where $a_n \in \mathbb{Z}$,$3 \nmid a_n $ My approach was to do it by ...
4
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2answers
137 views

Function $F$ is surjective if and only if $F$ is $1-1$ [duplicate]

While I was working on proofs of functions, the following claim occurred to me that I think it is correct but I could not prove it. Please note that the claim may not be correct since it is just my ...
1
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1answer
30 views

Equivalence with measurable functions and spaces

i have been reading Follands Real Analysis Book and i got stuck with one exercise. It says that if $\lambda(X)$ is finite and $(X,M,\lambda)$ is a measure space and $(X, \overline{M}, \overline{\...
1
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1answer
32 views

$\sum\binom{n}{a_1}\prod\limits_{p=2}^{k}\binom{n-a_{p-1}}{a_p-a_{p-1}}=k!{n+1\brace k+1}$

We have $$\sum\limits_{0\leqslant a_1<a_2<\cdots<a_k<n}\binom{n}{a_1}\prod\limits_{p=2}^{k}\binom{n-a_{p-1}}{a_p-a_{p-1}}=k!{n+1\brace k+1}$$ where instead of $\{a_1,a_2,\cdots,a_k\}$ we ...
1
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1answer
10 views

Question on Partially ordered sets and images of sets

Could someone look through my attempt at proving the following problem please? Let $(A,\preceq)$ and $(B,\preceq')$ be POSETS and $C \subseteq A$. Suppose that $h:A \rightarrow B$ satisifies $x \...
2
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2answers
38 views

Question regarding partially ordered sets and the subset relation

Could someone look through my attempt at proving the following problem please? Let $(A, \preceq)$ be a POSET.For each $x,y \in A$,let $P_x=\{a \in A: a \preceq x\}$. Let $F$ be the family of sets ...
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2answers
26 views

Sequence\Limits Proof

Let $L = \lim_{k \rightarrow \infty}\limits x_k$. If $(x_k)_{k=0}^\infty$ is increasing, then $x_k \le L$ for all $k \ge 0$ Could anybody push me in the right direction? I've stared at this one for ...
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1answer
13 views

defining a function. question about domain and how to specify validity

context: I'm working on a proof. I'm also working to improve my mathematical writing. questions Q.1: Is there a more compact, yet formal way of writing the following? ``Let $f\colon \mathbb {R} \...
0
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1answer
18 views

Question about Partially Ordered Sets and functions

Could someone verify my attempt at the following problem? Let $(A,\preceq)$ and $(B, \preceq ')$ be Partially ordered sets and suppose that $h:A \rightarrow B$ satisfies $x \preceq y \iff h(x) \...
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1answer
27 views

Prove the statement, For all integers $n ≥ 0$, $y_n = 3 * 2^n + 4^n$ where $y_0 = 4, y_1 = 10$ and when $n ≥ 2$, $y_n = 6y_{n-1} – 8y_{n-2} $

So approaching this problem For all integers $n ≥ 0$, $y_n = 3 * 2^n + 4^n$ where $y_0 = 4, y_1 = 10$ and when $n ≥ 2$, $y_n = 6y_{n-1} – 8y_{n-2} $ I realise that is probobly needs to be proved ...
8
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1answer
47 views

Is this proof of $H\le G, [G:H] =2 \implies a^2\in H \forall a\in G$ correct?

If $H$ is a subgroup of a group $G$ and the index (number of right cosets) of $H$ is $2$, then $a^2 \in H$ for all $a\in G$. My attempt: if $a\in H$ then $a^2\in H$ directly. If $a\notin H$ and $a^2\...
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0answers
14 views

Proof: matrix representation of a symmetric bilinear form over a $\mathbb{F}_q$-vector space

Consider $f$ a symmetric bilinear form over a finite-dimensional $\mathbb{F}_q$-vector space $V$, with $q$ odd. Then there exists a basis of $V$ such that the matrix $A_f$ of $f$ (with respect to that ...
1
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1answer
65 views

Let $P ⊆ X ×Y$. Does $π_1(P)×π_2(P) = P$? Give a proof or a counterexample.

Proofs and fundamentals, exercise 4.2.5. I need your help, maybe it's false. Let $X$ and $Y$ be sets, let $A ⊆ X$ and $B ⊆ Y$ be subsets and let $π_1 : X × Y → X$ and $π_2 : X × Y → Y$ be projection ...
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0answers
27 views

Show that all functions $X : \Omega \longrightarrow \mathbb{R}$ defined in a probability space is a random variable.

Show that all functions $X : \Omega \longrightarrow \mathbb{R}$ defined in a discrete probability space is a random variable. $\Omega$ is a sample space Answer: Because the definition of a random ...
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2answers
144 views

If R is an equivalence relation, and S is only symmetric and transitive, what is R ∪ S?

I have a question that asks the following: Let R and S be binary relations on a set A. Suppose that R is reflexive, symmetric, and transitive and that S is symmetric, and transitive but is not ...
1
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0answers
24 views

Question on modular arithmetic and well defined functions

Could someone help correct, if necessary, my attempt at proving the following question ? Let $\Bbb{Z}_3=\{[0]_3,[1]_3,[2]_3\}$ and $\Bbb{Z}_6=\{[0]_6,[1]_6,[2]_6,[3]_6,[4]_6,[5]_6\}$ be ...
1
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2answers
45 views

Prove that $\overline {B^*} = B.$

Let $R$ be a commutative ring with identity. Let $\operatorname {Max} (R)$ denote the set of all maximal ideals of $R$ and let $\operatorname {J} (R)$ denote the set of all prime ideals of $R$ which ...
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votes
4answers
123 views

proving that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$ [closed]

Knowing that $a+b=c+d$ and $a^3+b^3=c^3+d^3$, prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$. Can you guys please help me complete this proof, as I was attempting it yesterday, without getting far. ...
0
votes
1answer
54 views

Proving the independence of an equation, from certain variables [closed]

The different and unequal to zero real numbers x, y, z satisfy the equation $$x^3+y^3+m(x+y)=y^3+z^3+m(y+z)=z^3+x^3+m(z+x).$$ Prove that $$K=\left(\frac{x-y}{z}+\frac{y-z}{x}+\frac{z-x}{y}\right)\...
0
votes
1answer
75 views

My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

I'm trying to understand Galois theory and any help on this question I'm working on would be very much appreciated. Let $E$ be the splitting field of $x^3-5$ over $\Bbb Q$. Compute $\mathrm{Gal}(E:\...
0
votes
1answer
59 views

How to start this proof? The line through $(a,b)$ and $(c,d)$ is the best linear approximation of any function through those points, as $a\to c$.

I want to prove that the line passing through $(a,b)$ and $(c,d)$ is the best linear approximation of any function passing through those 2 points, as $a$ approaches $c$. I'm not sure how to define "...
0
votes
1answer
32 views

Prove the following by deduction rules for qualifiers and implication

I am having trouble with the following proof. First, some basic definitions: Def. Given two statements $\alpha$ and $\beta$, the statement \begin{equation}\alpha \implies \beta ,\end{equation} read ...
1
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1answer
82 views

If the proof is correct, is there a shorter way?

I came up with this proof for the inequality. If the proof is correct, is it possible to come up with a shorter one? I feel like my proof is very inelegant and there is a short trick that can be ...
2
votes
1answer
79 views

How does one subtract from concatenation in order to prove that $4\times 5 + 67 = 45 + 6\times 7$?

I noticed that if we get the numbers $4$, $5$, $6$ and $7$, they have an interesting property! $$4 \times 5 + 67 = 45 + 6 \times 7\tag*{= 87.}$$ I then conjectured that these were the only four ...
0
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0answers
74 views

the suprema of $p$ norms in a $n$-ball

CONTEXT I'm writing a proof. I'm trying to make sure my mathematical writing is correct. I want verify that the supremum $$\sup\left\{\left\|\textbf{x}\right\|_p \mid \textbf{x} \in B_r{\left(\bf{...