For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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7
votes
1answer
127 views

Nonlinear partial differential equations with applications

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
1
vote
3answers
55 views

$\forall x,y\in \mathbb N\ ,\ \exists\ z\in\mathbb N$ that $x+z$ is square but $y+z$ is not square.

I am trying to prove this: $\forall x,y\in \mathbb N$ and $x\neq y,\ \exists\ z\in\mathbb N$ that $x+z$ is square but $y+z$ is not square. $\mathbb N$ is set of natural numbers. Can you ...
2
votes
0answers
78 views
+200

Prove that in a group iterated commutators with repeated generators is trivial implies that each generator commutes with all its conjugates

Let $G$ be a finitely generated group with generating set $S=\{x_1,\cdots,x_n\}$. Let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Suppose that every iterated commutator with repeated ...
2
votes
1answer
56 views

How can a proof by formula induction in a formal language be formalized?

From a set of not-so-rigorous lecture notes on Metalogic: Formulas of $L$: (i) Each sentence letter is a formula. (ii) If $A$ is a formula, then so is $\neg A$. (iii) If $A$ and $B$ ...
2
votes
1answer
33 views

Graph Theory Proof (on connectivity)

Let G = (V, E) be a graph Provide a proof that shows G is connected iff there exists a walk that passes through every vertex in G. I understand that since it's a iff statement, there should be two ...
0
votes
0answers
24 views

Supposedly one can obtain the Jacobian from Stoke's theorem

My text uses geometric notions to convince us of the validity of the Jacobian in its usage for change-of-variables. I'm told that a proof for the 3D Jacobian can be obtained from Stoke's theorem. ...
0
votes
1answer
10 views

Powers inequality proof

I don't even understand what this proof is asking, let alone how to do it. here it is: Show that if $x>1$ is a real number and if $a<b$ are rational numbers, then $0\le x^a \le x^b$. any hints ...
0
votes
1answer
42 views

Multiplicative Identity is Unique

I'm having issues proving that the multiplicative identity is unique on the integers. Heres what I have so far, EDIT: Suppose $\exists \ \theta_{1},\theta_{2} \ such \ that \ \theta_{1} \neq ...
0
votes
1answer
13 views

Using Eigenvalues to prove a matrix?

In regard to eigenvalues and eigenvectors in Linear Algebra, How can I prove that the characteristic equation of a $2 \times 2$ matrix $A$ can be expressed as $$ \lambda^2- tr(A)\lambda + \det(A)=0 ...
0
votes
1answer
27 views

Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
2
votes
4answers
56 views

In a limit proof, what are the assumptions?

In a proof. Prove that given: $$\lim_{x \to a} f(x) = L$$ then $$\lim_{x\to a} |f(x)| = |L|$$ We know that $$|f(x) - L| < \epsilon \space \text{for} \space |x - a| < \delta_1$$ What is the ...
3
votes
1answer
26 views

Proof: Characterize m

Characterize $m$, an integer, such that $m^2≡1 \pmod{5}$. State your characterization as an "if and only if" statement and then prove it. This question is on my study guide for a test that is on ...
0
votes
2answers
27 views

Innovation behind formula for surface area and volume of a sphere

When I saw some problems about innovation behind area of a circle in this site,I was wondering that about a sphere.we know volume of sphere is $\frac{4}{3}\pi*r^3$ and surface area is $4\pi*r^2$,but ...
0
votes
0answers
19 views

Sum of minimums in set, relative to the largest element.

If $\mathbb{A}$ is finite set of non-negative numbers with cardinality equal to $n$. Sum of $\mathbb{A}$'s elements is $1$ prove inequality. $$\mathbb{A}_{\max} \cdot \left(1 + 2 \cdot ...
1
vote
1answer
36 views

Showing that $A \subseteq B$ for $A=\{6t\mid t \in \mathrm Z\}$ and $B=\{3t\mid t \in \mathrm Z\}$

Let $A=\{6t\mid t \in \mathrm Z\}$, and $B=\{3t\mid t \in \mathrm Z\}$. Then, show $A$ is a subset of $B$ and prove or disprove that $A = B$. I already know that $A \neq B$, for I can pick a ...
5
votes
1answer
116 views

Equivalence of two relations in Braid groups

Let $B_n$ be the braid group; that is, a group generated by $\sigma_1,\cdots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots,n-2$; ...
1
vote
2answers
48 views

differentiability of $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$

Let $f(x,y)=xy\sin\left({1\over x^2+y^2}\right)$ if $(x,y)\neq (0,0)$ and $0$ if $(x,y)=(0,0)$. Determine the points in which $f$ is differentiable I know that $f(x,y)$ is differentiable at ...
2
votes
2answers
37 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
1
vote
1answer
74 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If it will be cut diagonally, how many pieces will be splitted? If knife pass exactly by concatenating we assume there is no damage ...
0
votes
0answers
19 views

Make a conjecture about a closed form

There's a formula given $$ T(m,n) = \left\{ \begin{array}{l l} 1 & \quad \text{if $m=0$}\\ 1+T(n \mod m, m) & \quad \text{if $m>0$} \end{array} \right.$$ We're told to use ...
-2
votes
1answer
37 views

Corollary from Maximum Modulus Principle and Schwarz's Lemma

Need to prove this implication derived from the maximum principle, but have no clue how. $$\forall k=0,...,N. \ f^{(k)}(0)=0 \implies\exists M=const . \ \forall z.|z|\lt1:|f(z)|\le M|z|^{N+1}$$
0
votes
0answers
37 views

Derive the p-test if p is an irrational number

I would like some help with this problem. I've been given that we have a $k^p$ s.t. $p$ is irrational and $k$ is a positive integer satisfies the following: $k^r\lt k^p$ if $r$ is positive and ...
1
vote
7answers
130 views

Prove there exists a real number x such that

I'm not really sure how to get started on these problems. In class I learned that I can prove a statement by: proving the contrapositive, proof by contradiction, or proof by cases. There ...
0
votes
2answers
22 views

Proving intersection of sets with division of natural numbers

How can I find and prove I'm right, the intersection of sets $A_n = \{\frac{n}{1},\frac{n}{2},\frac{n}{3},\dots\}$ from $n=0$ to $\infty$ where $n \in \mathbb{N}$? It seems to me that the ...
1
vote
2answers
56 views

Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ by Induction

The Question Prove $1 + \frac{1}{4} + \frac{1}{9} + \cdots + \frac{1}{n^2} < 2 - \frac{1}{n}$ where $n\ge2$ and $n$ is an integer by Induction My Work Basis Step: 1 + $\frac{1}{4} = ...
1
vote
3answers
39 views

Show that $n!<n^n $ where $n>1$ and is a Positive Integer

Basis Case: $2! = 2\times1 = 2$ $2^2 = 4>2$ Inductive Hypothesis: $k!<k^k$ Induction Step: $k!<k^k$ $k!(k+1) < k^k(k+1)$ $(k+1)! < k^{k+1} + k^k$ I'm confused on where to go ...
2
votes
0answers
28 views

could someone help me with this particular detail in this article?

I'm reading this article and I was stuck in this part: I didn't understand his trick computing the orders at $P$. Remark: He defines $D=\inf\{div(\omega_{g-1}),div(\omega_g)\}$, i.e., ...
1
vote
1answer
26 views

Proving Lower Bound on Catalan Numbers

I'm a student of computer science and was reading through my algorithms textbook about matrix chain multiplication. It brought up Catalan numbers and I was hoping to prove the lower bounds on it. This ...
2
votes
2answers
314 views

Counterexample to if g ◦ f is surjective, then f and g are surjective

I want less of an answer and more of an explanation if possible please. I understand I'm looking for a range of either f and g that is NOT surjective(does not cover all the codomain), but that their ...
0
votes
0answers
26 views

Trying to prove any symmetrical matrices would be a vector space

Here is the question I am struggling with; Let $V$ be the set of any real symmetric matrices, that is, the set of all matrices $A$ such that $A^{T}=A$ For whatever reason I just can't seem to find ...
0
votes
0answers
25 views

Define f:Z/3Z→Z/3Z by f([a])=[2a+1]

Just finished proving this to be injective, and well-defined. How would you prove it to be surjective? I understand surjective means that every element in the codomain is being used, and thus is the ...
2
votes
1answer
25 views

Define $f : Z/3Z → Z/3Z$ by $f ([a]) = [2a + 1]$

Just finished proving this is well-defined, how do I prove it's surjective and injective? I know that injective means that if $x1 \neq x2$, then $f(x_1) \neq f(x2)$, i.e. each value in the domain is ...
0
votes
1answer
22 views

Define f: Z/4Z → Z/4Z by f([a]) = [3a+1]

I need to show this function is well defined For well defined, I was thinking something along the lines of: Assume [a1] = [a2] in Z/4Z. Then, a1 is congruent to a2(mod4). So, 4 | a1 - a2. Thus, 4 | ...
0
votes
2answers
22 views

How do I prove $e^z$ is a covering map using this fact?

I have proven that $p:\mathbb{R}\rightarrow S^1:t\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a covering map and $S^1$ and $\mathbb{C}\setminus\{0\}$ are homotopically equivalent. Using these facts, how do ...
0
votes
1answer
24 views

Z / 6Z being a set of well dedfined equivalence classes, and a congruent to b(mod 6)

why is this = [0],[1],[2],[3],[4],[5],[6] and how would I define f Z/6Z - Z/6Z by f([a]) = ([2a]). I have the proof but I don't understand it. Proof: Assume [a1] = [a2] in Z/6Z. then a1 congruent to ...
1
vote
1answer
22 views

How to prove by this type of question by Induction (If $a_1 = 6$ and $a_{m+1} = 2a_m - 3m + 2$ for $m \geq 1$, then $a_n = 2^n + 3n + 1$)

Please do not tell me how to prove this exact question. I would like to know how to go about proving the following type of question by induction: If $a_1 = 6$ and $a_{m+1} = 2a_m - 3m + 2$ for $m ...
0
votes
1answer
15 views

Define f: Z /6Z by g(5[a]) = [5a]

So, in our notes, we had an example where we defined f: Z / 6 Z by g([a]) = [5a] (where z is set of all integers) Already, I don't follow what the g([a]) = [5a] means, I'm assuming they are ...
12
votes
1answer
584 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
1
vote
0answers
20 views

Prove or disprove: $L^2$ context free implies $L$ is context free.

Clearly we have to disprove this. But I am finding it hard to prove it. I was trying in following way: Considering any non context free language $L$. I was trying to prove that $L^2$ is context free ...
0
votes
1answer
34 views

Could someone point me in the right direction for this proof?

I need to prove that Q (rational numbers) is countable by applying the function $f(m/n) = 2^m3^n$ with $m,n$ being relatively prime numbers. I honestly have no idea where to start. Any pointers ...
1
vote
1answer
63 views

Something I don't understand about Hilbert's grand hotel

So I want to know if Hilbert's hotel "story" holds for this statement: $\wp (\mathbb{N}) \sim \wp (\mathbb{N})\smallsetminus \left \lbrace\emptyset\right\rbrace$ So, If the statement wasn't talking ...
1
vote
3answers
23 views

Confused about limit proofs conceptually

In a question like this: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x), g(x)) = \max(l, m)$ In general, when asked for proofs like this, are ...
-1
votes
1answer
25 views

How to prove that cardinal numbers of sets and unions of them are equal

Let A, B, C, D be sets. If I know that $A\sim C$ and that $B\sim D$, In addition I know that: $C\cap D = \varnothing$ and also, $A\cap B = \varnothing$ Does it imply that $A\cup B\sim C\cup D$? ...
2
votes
3answers
100 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
0
votes
0answers
31 views

Proof of Correctness: Recursion inside loop

I am trying to prove the correctness of the algorithm in the research paper. It is at page 17 in the pdf. ...
1
vote
2answers
43 views

Proof of Bézout's identity - Cohn - CA p26

Given two integers $a$ and $b$, there exist integers $u$ and $v$ such that $$au+bv=1$$ if and only if $a$ and $b$ are coprime. Attempt Proof: Assume $a$ and $b$ are not coprime, e.g. $a=kb,k\in ...
1
vote
2answers
51 views

Maximum and minimum function on an interval

Let $I := [a,b]$, where $a<b$. Suppose that $f$ is continuous and $1-1$ on $I$. Let $m$ denote the minimum value of $f$ on $I$ and let $M$ denote the maximum value of $f$ on $I$. (a) Carefully ...
0
votes
1answer
87 views

Using the contraction principle to prove that the sequence $a_n=f(a_{n-1})$ is convergent where $f(x)=1+1/x$

Define function $f(x)$ from $\mathbb R^1$ to $\mathbb R^1$ by $f(x) = 1+1/x$. Define $a_n$ inductively by $a_1=1$ and $a_n=f(a_{n-1})$ for $n \geq 2$. Prove using the contraction principle that ...
0
votes
1answer
44 views

Convergent Series 2n-1/2n

Prove the series defined by P(n) = (1 *3 * 5 * (2n-1))/(2*4*6 * (2n)) is convergent It is monotone decreasing and bounded below by zero, but is that enough to say?
0
votes
2answers
24 views

Prove by induction fibonacci variation

Prove by induction: The fibonacci sequence is defined as follows: $f_1 = 1$, $f_2 = 1$ and $f_{n+2} = f_n + f_{n+1}$ for $n \geq 1$ Prove by induction that $f_1^2 + f_2^2 + \dotsb + f_n^2 = f_n ...