Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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19 views

Maximum $n$ such that $e2^{1 - {k \choose 2}}{k \choose 2}{{n-2} \choose {k-2}} \leq 1$, as $k \to \infty$.

Using: ${k \choose 2} \leq \frac{k^2}{2}$, and ${{n-2} \choose {k-2}} \leq \frac{k^2}{n^2}{n \choose k} \leq \frac{k^2}{n^2} \left(\frac{ne}{k}\right)^k$, we have $e2^{1 - {k \choose 2}}{k \choose ...
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1answer
42 views

Countability of the set $(0,1)$

I am trying to prove that the set $(0,1)$ is uncountable from "A First Course in Analysis by Yau". I have a question about a particular step. In the text, the result is proved by contradiction. It is ...
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1answer
19 views

For $G \sim G(n, 0.5)$, $I$ a $k$-set of vertices, and event $E_I = \{G[I] \cong K_k \text{ or } K_k^c \}$, how many events does $E_I$ depend on?

The note in my lecture says that $E_I$ is independent of all events with disjoint edge-sets, and so it depends on at most ${k \choose 2}\left({{n-2}\choose {k-2}}-1\right)$ other events. I don't ...
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1answer
25 views

Explain a section of Euclid's Theorem on an infinite number of prime numbers.

I'm trying to understand Euclid's Theorem, using proof by contradiction, which states: There are an infinite number of prime numbers. In the book it has the following explanation: We assume that ...
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1answer
25 views

Proof of uniqueness of Laplace Transform

I'm studying proof of the uniqueness of the Laplace Transform and I have some problems with understanding it. Here is the proof. Why can we change $u^{n}$ for polynomial $p(u)$? What does it mean?...
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25 views

m pages are printed, and the number of pages misprinted is never even. Why is any of the m-1 events $E_i =$ {misprint in the ith page} are m.i.?

("m.i." = "mutually independent") My understanding is that the conclusion is equivalent to saying that amongst a particular set of $m-1$ events, any one of those events would not depend on any subset ...
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1answer
46 views

Doubt regarding the proof of row rank = column rank

Wikipedia provides two methods to prove row rank of a matrix is equal to its column rank. My doubt is regarding the second method. But the wikipedia page mentions that this proof is valid only for ...
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1answer
24 views

Variance of $X$, an uniformly random sum from a finite set $S$. [closed]

This is from my class. Can I have an explanation of what going on in the last equality (i.e. $\operatorname{Var}\left(\varepsilon_{i}\right) s_{i}^{2}=\frac{1}{4} \sum_{i=1}^{k} s_{i}^{2}$)? ...
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2answers
59 views

Some doubts about proof of Strong Law of Large Numbers

I quote Jacod-Protter. Theorem: Let $\left(X_n\right)_{n\geq1}$ be independent and identically distributed and defined on the same space. Let$$\mu=\mathbb{E}\{X_j\}$$ $$\sigma^2=\sigma_{X_j}^2<\...
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1answer
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Proof for Division Algorithm, from the book *Contemporary Abstract Algebra* by Joseph A. Gallian

This is the same question as Q2 of Need help with understanding the proof for Division Algorithm, from the book *Contemporary Abstract Algebra* by Joseph A. Gallian "Q2: I don't understand how 0∉S ...
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1answer
23 views

Understanding the fulfillment of a condition required for applying the pasting lemma.

Here is the solution of the question (which asks us to prove that the relation of homotopy among maps $X \rightarrow Y$ is an equivalence relation): My question is about the last part of proving ...
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3answers
53 views

How is this function injective?

I'm currently studying Galois Theory and I came across this theorem. Theorem Let $E$ be a field, $p(x)\in E[x]$ an irreducible polynomial of degree $d$ and $I = \langle p(x) \rangle$ the ideal ...
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Rules when prooving left side with right side

I want to prove that some statement $A$ is equal to another statement $B$, i.e., $A = B$. I can start with the statement $A$ and try to reach $B$ and vice-versa, or I can start with $A$ and reach $C$...
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1answer
66 views

Doubt about series manipulation in proof of Abel's Limit Theorem

In my book a proof of Abel's Limit Theorem is presented. The theorem as stated: Let$f(x)$ be the sum function of the power series $\sum_{n=0}^\infty a_nx^n$, which has radius of convergence 1; and let ...
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3answers
55 views

If $\lambda \not= 0$ and $\lambda x = 0$, then $x = 0$

I am trying to use the definition of vector spaces to prove that, if $\lambda \not= 0$ and $\lambda x = 0$, then $x = 0$. One proof I have seen begins as follows: $\lambda 0 = 0$ for each $\...
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1answer
43 views

De Morgan’s law: Wikipedia proof, cannot follow part 1, step 3.

I would like to prove De Morgan’s laws and have tried to follow the Wikipedia proof. However, I am stuck in part 1 of this proof, line 3: 1: Let $x \in (A \cap B)^c $. Then, $ x \notin A \cap B $. 2:...
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1answer
39 views

prove that $ \frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n $

[HMMT 2004] For every positive integer $n$, prove that $ \frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n $ If $d$ is a divisor of $i,$ then so is $\frac{i}{d},$ and $\...
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1answer
70 views
+100

How can these two subsets be homeomorphic?

From Rotman's Algebraic Topology: Prove: If $K$ and $L$ are simplicial complexes and if there exists a homeomorphism $f: |K| \rightarrow |L|$, then $\text{dim }K = \text{dim } L$. Partial proof: ...
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1answer
33 views

Vector spaces uniqueness proof: If $z_1$ and $z_2$ are two such elements, then $z_1 + z_2 = z_1$ and $z_1 + z_2 = z_2$; thus, $z_1 = z_2$.

I am currently studying Introduction to Hilbert Spaces with Applications, by Debnath and Mikusinski. Chapter 1.2 Vector Spaces says the following: (a) $x + y = y + x$; (b) $(x + y) + z = x + (...
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1answer
30 views

Prove that $L \geq M$.

In the above proof, it is taken that for both of the sequences $\exists N$ s.t $|a_n - L|<\epsilon $ and $|b_n - M|<\epsilon$. Is it correct to take the same $N$ for both sequences? After that ...
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1answer
53 views

Comparing 2 solutions of problem 2 chapter 0 of Allen Hatcher.

The question is to construct an explicit deformation retraction of $\mathbb{R^n} - \{0\}$ onto $S^{n-1}.$ Here is the answers I found online so far: The first solution: The second solution ...
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1answer
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Proof Explanation: If $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$

Synopsis In Exercise 4.23 of Enderton's Elements of Set Theory, we are asked to show that if $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$. This seems like an obvious statement, but I ...
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1answer
23 views

Computation of the second derivative of the Jacobian of the change of the coordinates

I am trying to understand how to derive the equality $1.135$ of the book "A course in minimal surfaces" by Colding and Minicozzi. I derive $$(g^{ij}g_{ij}')' = (g^{ij})' g_{ij}' + g^{ij} g_{ij}''$$ ...
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23 views

Let $F$ be a face of a polyhedron $P$ and $P'$ the first Chvátal closure of $P$. Then $F' = P' \cap F$.

The above is a lemma from my class. I'm looking at this picture below to make sense of it. In the picture, I'm focusing on the face $F$ of $P$ defined by (1)(2)(4). I've learned that $P'$ is defined ...
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0answers
29 views

Let $f:[0,1] \rightarrow [0,1]$ continuous. Prove that $x\in [0,1]$ exists that $f(x) = x$ [duplicate]

MY TRY: So if the function is continuous it means: $$|x-a|<\delta$$ Because the function is bounded we already know that $\delta \in (x-1,x+1)$ or that $\delta \leq 1$ and $a \in [0,1]$ There ...
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1answer
21 views

Why $(n-1)S^2=σ^2(z_1^2+z_2^2+…+z_{n-1}^2)$?

This question is about the Wishart distribution. enter image description here I can do the $(n-1)\dfrac{Σ_{i=1}^n(x_i-x̄)^2}{n-1}=Σ_{i=1}^n(x_i-x̄)^2$ $=\dfrac{Σ_{i=1}^N(x_i-x̄)^2}{N}((\dfrac{x_1-...
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1answer
56 views

To prove that $p$ is a prime number

I'm reading a book about proofs and fundamentals on my own and, currently, I'm having trouble proving this result. Theorem: Let $p$ be a positive integer bigger than or equal to $2$ and such that, ...
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2answers
29 views

Substitution in diophantine equation

If I have an equation $$3(1+x+x^2)(1+y+y^2)+1=4x^2y^2 $$ and I am interested in non negative integer solutions. I let $x$ be the smaller of the positive integer solution so I substitute $x+\lambda=y$ ...
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2answers
82 views

Calculus proof of ln(ab)= lna + lnb

My calculus book states the following theorem of the properties of natural logarithms: If a, b > 0 , then ln(ab)= lna + lnb The author goes on to prove this theorem as follows I do not understand ...
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1answer
62 views
+50

Signed measure: verification

Let $f\colon X\to[-\infty,+\infty]$ a measurable function such that $$\int_Xf^+\;d\mu<\infty\quad\text{or}\quad \int_Xf^-\;d\mu<\infty,$$ then $$\nu(E)=\int_Ef\;d\mu$$ is a signed measure. I ...
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2answers
54 views

Proof verification and explanation in probability

Six regimental ties and nine dot ties are hung on a tie holder. Sergio takes two simultaneously and randomly. What is the probability that both ties are regimental? I have seen that the ...
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27 views

Suppose n>2 and a^(n-1)≡2 mod n then

Suppose $n>2$ and $a^{n-1}≡2$ (mod $n$) then $a$ and $n$ may not be relatively prime, so I cannot draw any conclusions. $n$ is not prime $a$ is not prime Which one is true?
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1answer
47 views

Olympiad Number theory Problem Solution Doubt

[Iberoamerican $1998]$ Let $\lambda$ be the positive root of the equation $t^{2}-1998 t-1=0 .$ Define the sequence $x_{0}, x_{1}, \ldots$ by setting $ x_{0}=1, \quad x_{n+1}=\left\lfloor\lambda x_{n}...
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1answer
38 views

Limit in Proving Brouwer's Fixed Point using Sperner's Lemma

I'm trying to understand the proof of Brouwer's Fixed Point Theorem using Sperner's Lemma. For example, pg.10-11 in A Combinatorial Approach to The Brouwer Fixed Point Theorem. I was able to follow ...
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1answer
38 views

Proving that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear

My lecturer presented a proof of the fact that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear. To show that the map is linear with $y$ fixed he wrote the following: $$\...
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1answer
14 views

Some doubts about Levy's Continuity Theorem proof - Convergence results

THEOREM (Levy's Continuity Theorem) Let $(\mu_n)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $(\hat{\mu}_n)_{n\geq1}$ denote their characteristic functions (or Fourier ...
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28 views

Finding a complement of $U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$

Consider the vector space $V=C(\mathbb{R},\mathbb{R})$ and $V\ni U=\{f\in C(\mathbb{R},\mathbb{R}) |f(0)=0\}$. I want to find a complement of $U$, such that $V=U\oplus W$. This condition is the same ...
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2answers
38 views

Determining if a function $f:C[0,1]\rightarrow M_{2,2}$ is surjective, injective or bijective

Determine if the function $f:C[0,1]\rightarrow M_{2,2}$ given below is surjective, injective or bijective: $$f(h)=\begin{pmatrix} h(0) & h(1) \\ h(1/2) & h(1)-h(0) \end{pmatrix}.$$ ...
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1answer
30 views

Proof of $\sup|f(x)|=\left \| f \right \|_\infty $

Let $(\mathbb{R}^n,L_n,\lambda _n)$ be a measure space and let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous and bounded, then $\sup\lvert f(x)\rvert = \lVert f \lVert_\infty $ . Proof: we ...
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27 views

Understanding Fraleigh's derivation of Fermat's Little Theorem

Here's how Fraleigh (informally) derives Fermat's Little Theorem in his classic Abstract Algebra text: For $\mathbb{Z}_p$, the elements $$1, 2, \dots, p-1$$ form a group of order $p-1$ under ...
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27 views

Prove that if MST T1 has k edges with weight 1 then T2 has also k edges with weight 1

We have two different minimum spanning trees ($T_1$ and $T_2$) of $G$. The graph $G$ has edges with weight $1$ or $2$. How can I prove that if $T_1$ has $k$ edges with weight equal to $1$ then $T_2$ ...
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1answer
23 views

What do first derivatives, factorials, and alternating signs have to do with explicit and recursive forms of sequences?

I'm a math teacher now, although a few years ago I was finishing up my M.Ed. As part of my studies, I was tasked with conducting my own study of high school level topics and finding unique results. As ...
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1answer
27 views

Projection on $U$ along $W$ with $U,W$ contained in $V=U\oplus W$

Assume $V=U\oplus W$ is a direct sum where $U$ and $W$ are subspaces of the vector space $V$. Then we can define a linear map $$E(v)=u,\\ with \qquad v=u+w\in V,\qquad u\in U,\ w\in W$$ Called the ...
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1answer
37 views

Doubt in a part of the proof of Bolzano-Weierstraß Theorem

Let $n$ be a natural number and $\{a_n\}$ be a bounded sequence of real numbers, that is $\vert a_n\vert\leq M$, for all $n$ ($M\geq0$). Define $E_n=\overline{\{a_j\vert j\geq n\}}$ as the closure of ...
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2answers
63 views

Prove that $p$ divides $ \left(1 ^ {p^ n} + 2 ^ {p^ n} +\cdots+ (p-1)^ {p^ n}\right) $. [duplicate]

Let $p > 2$ be an odd number and let $n$ be a positive integer. Prove that $p$ divides $ \left(1 ^ {p^ n} + 2 ^ {p^ n} +\cdots+ (p-1)^ {p^ n}\right) $. This is a question from Titu book on number ...
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2answers
17 views

Let $a$ & $b$ be non-zero vectors such that $a · b = 0$. Use geometric description of scalar product to show that

Let $a$ & $b$ be non-zero vectors such that $a \cdot b = 0$. Use geometric description of scalar product to show that $a$ & $b$ are perpendicular vectors. What I've done so far is to state ...
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1answer
36 views

How to Prove a Special Case of Stokes' Theorem?

I am currently in Calculus 3, or Multivariable Calculus and need to prove this special case of Stokes' theorem. Please forgive me as I do need this simplified to the bones to understand the ...
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1answer
15 views

sum of pairs and sum of squares $\left(\sum x_i\right)^2-2\sum_{cyc}x_ix_j=\sum x_i^2$

I was messing around today and came across the following which I believe to be true: $$\left(\sum x_i\right)^2-2\sum_{cyc}x_ix_j=\sum x_i^2$$ for a set of $x=(x_1,x_2,...x_n)$ where $n\ge2$ and it is ...
2
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2answers
52 views

Explanation of how to get $38x+19y=3xy$ into factorised form.

The core problem I would like explained is how to get: $38x+19y=3xy$ into factorised form (not necessarily equal to zero though). This is the method proposed in my book: $3xy-19y-38x=0$ ...
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0answers
26 views

Why are there $\mathcal{O}(\dfrac{n}{\log n})$ prime divisors of value $\Theta(n^c), c>4$ wich divide a number $\leq 2^n$? [closed]

I am trying to understand this very small proof in in this picture from "A Fully Dynamic Algorithm for Maintaining the Transitive Closure" (https://www.sciencedirect.com/science/article/pii/...

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