For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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2
votes
1answer
63 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
2
votes
3answers
79 views

Relating geometric and Algebraic Definitions of the dot product

I am about to enter the class Engineering Physics II. Alas, much of my mastery of vector manipulation is predicated on something I don't understand that must be taken as an assumption for me to ...
1
vote
1answer
96 views

Find the index of the equilibrium points of the system (Question on solution)

I have the following system: $$\dot{x} = 2xy$$ $$\dot{y} = 3x^2-y^2$$ I have the following solution: The system has one equilibrium point at the origin. Let the curve $\Gamma$ surrounding the origin ...
1
vote
2answers
74 views

EDIT: Proving $f^{-1}(f(C))=C$

I need to prove that $f^{-1}(f(C))=C$. This are the informations. There exists two sets A and B, and function $f(A)\to B$. I don't know how to solve this, and I tried to search google, but I didn't ...
1
vote
3answers
160 views

Proof that $\frac{(x+y)-abs(x-y)}{2}$ equivalent to $\min(x,y)$

I plotted the two functions $\frac{(x+y)-abs(x-y)}{2}$ and $min(x,y)$ in the range $[-1, 1]$ and they look the same. The both $min$ and $abs$ functions are defined as expected. ...
1
vote
1answer
166 views

How to linearize a nonlinear ODE around its equilibrium?

I am studying for a comprehensive exam in non-linear ODE's and I have this in my book: $$\ddot{\xi}+c\bigg[x_1+\xi-\dfrac{\lambda}{a-x_1-\xi}\bigg] = 0$$ then it goes straight to ...
3
votes
0answers
67 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
1
vote
1answer
37 views

Qustion about field and sub-group

$F$ is a field and and $H$ is finite sub-group of $(F,\cdot)$ ($F$ without the $0_F$). I need to prove that $H$ is cyclic. I can use this fact - Can we conclude that this group is cyclic?. (I don't ...
0
votes
1answer
65 views

Proof adding layers of constant width to a shape tends to an $d$-sphere as the number of layers tends to $\infty$

Good night, I've recently seen one of Victoria Hart's videos on Youtube (it wasn't about this, it was about Fibonacci numbers, and I found it on a comment in this site), and in it she said that if ...
1
vote
1answer
168 views

prove Turing recognizable

This is actually an old exam question its not my homework; Let L = { : M is a TM with an input alphabet of {a,b} and M accepts at most one word, i.e. M either accepts no words or accepts exactly one ...
0
votes
1answer
22 views

what conditions to take while proving a result.

Suppose I have a theorem's statement as follows: If statement A and statement B, then statement C. I want to prove the converse, but quite confused what conditions to consider. I got hint as ...
3
votes
3answers
250 views

Premise vs. Assumption

I have just asked about the difference between A,B and A∧B in A,B ⊢ M However, I have ...
1
vote
3answers
84 views

Bounded Quantifiers in a Proof

In a solutions manual for Jech's "Introduction to Set Theory", we find the following proof for $f\left[ \bigcap_{a\in A} F_a \right]\subseteq \bigcap_{a\in A}f\left[ F_{a}\right]$. \begin{align} y\in ...
1
vote
2answers
123 views

If $2$ divides $p^2$, how does it imply $2$ divides $p$?

I'm trying to understand a proof by contradiction. It's proving by contradiction that $\sqrt2$ isn't rational. (It's a standard proof involving $\sqrt2=\frac{p}{q}$, where $p,q$ are already ...
12
votes
2answers
363 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
2
votes
0answers
78 views

If $a\lt 0$ and $b\lt 0,$ then $ab\gt0$.

$\quad$The following assertion is somewhat less obvious: If $a\lt 0$ and $b\lt 0,$ then $ab\gt0$. The only difficulty presented by the proof is unraveling of definitions. The symbol $a\lt 0$ means, ...
2
votes
0answers
163 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
vote
1answer
79 views

If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some $k\in\mathbb N$.

I would like to know if my proof below is correct. Problem If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$. ...
3
votes
3answers
137 views

If $\Omega = \{1,2,3,\ldots,\}$, then $S_{\Omega}$ is an infinite group.

I would like to know if my proof below is correct. I do not have issues proving that $S_{\Omega}$ is a group; what I am not sure is whether my proof that $\vert S_{\Omega} \vert = \infty$ is correct. ...
2
votes
0answers
100 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
4
votes
2answers
38 views

Formatting for problem sets?

What is considered a good format for writing problem sets in mathematics? Are there any good examples of problem sets that are well-written and formatted that you can show me?
0
votes
1answer
35 views

Question about question about finite order elements at $\mathbb{C}^*/U$

At this question - What are element with finite order at $\mathbb{C}^*/U$? I understand that finite order at $\mathbb{C}^*/U$ are only the $e$ elements. Now, I have two questions: It is because ...
1
vote
1answer
50 views

Prove thant if $a/b + c/d \in \mathbb Z, (a:b)= 1, (c:d) =1 $ then $|b|=|d|$

Be $a,b,c,d \in \mathbb Z, b \ne 0, d \ne 0.$ Prove that if $a/b + c/d \in \mathbb Z, (a:b)= 1, (c:d) =1 $ then $|b|=|d|$
1
vote
4answers
67 views

Proof that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}$

Being $\mathbb G_n$ the roots of unity for $n \in \mathbb N$, prove that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}.$
1
vote
1answer
61 views

Given $A$ and $\vec{b}$ in $A\vec{x}=\vec{b}$, solve for $\vec{x}$

What are the steps to solve for $\vec{x}$, given that $A\vec{x}=\vec{b}$ and we know what $A$ and $\vec{b}$ are? I know the first thing you do is multiply each side by $A^T$ ...
1
vote
2answers
251 views

Prove that the sum of two positive integers is positive? [closed]

On a practice final exam for my Discrete Math class, I've been asked to prove that the sum of two positive integers is positive. I've been pulling my hair out over how to prove this, as it seems so ...
3
votes
3answers
79 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
2
votes
1answer
28 views

Proof of proportions in proportion by composition and decomposition?

What is the proof that $\frac{a+b}{a-b}=\frac{c+d}{c-d}$ given that $\frac{a}{b}=\frac{c}{d}$ Here's what I've got so far: $$\begin{array}{l} \text{Statements} ...
1
vote
0answers
58 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
3
votes
2answers
58 views

What are element with finite order at $\mathbb{C}^*/U$?

I need to find the elements with finite order at the group - $\mathbb{C}^*/U$. $U$ - is the Circle Uint. $\mathbb{C}^*$ - is $(\mathbb{C}/0,\cdot)$. I need to write also the proof, and I'll be glad ...
3
votes
2answers
86 views

Is this approach to induction valid?

This is a homework problem: Prove that: $$ 3^{4n+1} + 5^{2n+1}$$ is divisible by $8$ for every natural number $n$. Base case: $$n = 0$$ $$ 3^{0 + 1} + 5^{0 + 1} = 8$$ $$8\bmod8 = 0 $$ Base case ...
2
votes
3answers
72 views

What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don't know how to prove it... I'd like to get help with the proof ...
1
vote
2answers
96 views

What software is useful for generating diagrams to use in formal proofs?

What software is useful for generating diagrams to use in formal proofs? I am interested in software for geometry diagrams, graphs, plots, and any other useful kinds of diagrams.
13
votes
4answers
362 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
0
votes
3answers
113 views

Is this true? $(1+1/n)^n=1+1/1!+1/2!+1/3!+1/4!+\cdots + 1/n!$

Is this true? $$\left(1+\frac{1}{n}\right)^n=1+1/1!+1/2!+1/3!+1/4!+1/5!+\cdots $$($n$ times) or ($n+1$ times)? If yes how to prove it and were there any proof of it?
0
votes
2answers
85 views

Verification of Proof of a Bijection from A to B

Problem: For $ a,b \in \textbf{R}$ with $ a < b$, prove an explicit bijection of $ A = \{ x : a < x < b \} $ onto $ B = \{ y : 0 < y < 1\} $. My attempt: We consider $ f(x) = ...
0
votes
2answers
37 views

Prove $(\forall n \in \Bbb N)[\gcd\left(n,(16n+1)^3\right)=1]$

Prove $(\forall n \in \Bbb N)[\gcd(n,(16n+1)^3)=1]$ Knowing that $\gcd(a,b)=\gcd(a,b+a\times k)$ with $k \in \Bbb Z$ $$\gcd\left(n,(16n+1)^3\right)=\gcd\left((16n+1)^3,n\right)=d$$ ...
1
vote
0answers
48 views

Proof of the Surfer Model Pagerank formula

How do you prove this formula for the Surfer Pagerank algorithm mathematically? ...
0
votes
1answer
49 views

Verification of proof that $f(x) = \frac{x-a}{b-a}$ is bijective over the reals

We consider $ f(x) = \displaystyle \frac{x-a}{b-a} $ for $f: \textbf{R} \rightarrow \textbf{R} $ where $a,b$ are both constants such that $a,b \in \textbf{R} $ and $b-a \neq 0$. Proof that $ f$ is ...
1
vote
1answer
56 views

Method for proving $ f^{-1}(G \cup H) = f^{-1}(G)\cup f^{-1}(H) $

Prove that if $ f: A \rightarrow B $ and $ G,H $ are subsets of $ B $, then $ f^{-1}(G \cup H) = f^{-1}(G)\cup f^{-1}(H) $. My (incorrect) Attempt: Suppose $ x \in f^{-1}(G\cup H) $. Then there ...
0
votes
1answer
31 views

Defining substitution by structural recursion

For a term u, let $u{x\atop t}$ be the expression obtained from $u$ by replacing the variable $x$ by the term $t$. Define $u{x\atop t}$ by recursion on $u$. Not really sure how to start this one. ...
0
votes
2answers
47 views

Problem proving connected subspace of complex plane touches real line

I've stated the problem hereunder and my ideas, but I feel that I'm missing something in my proof. I'm not sure that my argument of "forcing connection" is actually a proof that my 4 conditions really ...
0
votes
1answer
110 views

Useful maths symbols - tips and made up symbols for quick writing?

I am looking for some symbols for the most common words in maths problems, such as : such as (":"), given (for some reason I am using $\sqsupset$ ), assume (I use $\downarrow$ ) etc and any common ...
0
votes
1answer
77 views

Simple Proof question

Image : http://postimg.org/image/dkn0d5uen/ I'm studying Spivak's calculus and I have a really simple question : I'm only in the first chapter on "The basic properties of numbers" So far, we have ...
1
vote
1answer
84 views

Is this a correct proof by contradiction?

Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then $A$ contains all natural numbers $\geq n_0$. I have attempted a proof by contradiction as ...
2
votes
2answers
78 views

Proof that $\gcd$ divides $\operatorname{lcm}$

Show that the following conditions are equivalent: i) There exist positive integers $a, b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$. ii) $d\mid m$
1
vote
3answers
123 views

How do I derive $(\forall x)(\forall y)(\exists z)(x = y \circ z)$ from these three group axioms and some previously established theorems?

I am currently self-studying Patrick Suppes' Introduction to Logic and I am stuck on exercise 5.2.4. I've successfully worked out proofs for Theorems 1 to 7, but I am having trouble coming up with a ...
1
vote
4answers
253 views

Prove by induction that $10^n -1$ is divisible by 11 for every even natural number

Prove by induction that $10^n -1$ is divisible by 11 for every even natural number n. $0 \notin N$ Base Case: n = 2, since it is the first even natural number. $10^2 -1 = 99$ which is divisible by ...
3
votes
1answer
54 views

How can I formalise a proof of this?

A question from a textbook: Take the interval $(a,b)$, split into thirds likewise: $$(a,b)=\Big(a,\tfrac{1}{3} (2a+b)\Big)\bigcup \Big[\tfrac13(2a+b),\tfrac13 (2b+a)\Big]\bigcup ...
2
votes
1answer
204 views

Discrete Math Proof By Cases Confusion

I am currently finishing up my Discrete Math course, and I just wanted to clear something up that has confused me for the past few days. My teacher posts answer keys to assigned homework problems ...