# Tagged Questions

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.

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, ...
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 ...
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 ...
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 $B$ ...
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 ...
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. ...
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 ...
42 views

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 ...
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., ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 | ...
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 ...
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 ...
22 views

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 ...
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 ...
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 ...