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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1answer
40 views

Proof by induction that if $a_0 = 1$ and $a_n = n + 2 a_{n-1}$, then $a_n \ge 2^n + n^2$.

I have that $a_0 = 1$ and $a_n = n + 2 a_{n-1}$ for $n \geq 1$. Now I need to proof by induction that $a_n \geq 2^n + n^2$. I already have my base case. My hypothesis would be $a_{n-1} \geq 2^{n-1} ...
0
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1answer
40 views

Assume G is a group, x,y is in G; x and y are not identity, but $x^3=1$ and $y^2=1$ and $(xy)^2=1$. Find the order of G and the group table

So I am stuck with this problem and I can't seem to find the relationship with the x, y and identity in dealing with size of group and how they connect with $(xy)^2=1$. Can someone help me with this? ...
0
votes
2answers
45 views

Prove $\forall a,b \in \mathbb{Z}, 18a + 6b \ne 1$

Prove $\forall a,b \in \mathbb{Z}, 18a + 6b \ne 1$ Is there a way to do this using proof by contradiction without using mod?
1
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3answers
56 views

Prove $\forall a,b \in \mathbb{Z}$, $a^2 -4b - 3 \ne 0$ using proof by contradiction

Prove $\forall a,b \in \mathbb{Z}$, $a^2 -4b - 3 \ne 0$ I want to do a proof by Contradiction. I know that this can be figured out using Rational root theorem by subbing in (1, -1, 3, -3), but I am ...
0
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0answers
42 views

How can I prove this theorem?

Let n ∈ N. Let b ∈ Z. Then there exists c ∈ Z satisfying c ·n b = 1
1
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1answer
26 views

Identity Tranformation Proof- Is this enough to prove this statement?

Let {v$_1$,...,v$_n$} be a basis for a vector space V and let T:V$\to$V be a linear transformation. Prove that if T(v$_1$)= v$_1$,...,T(v$_n$)= v$_n$, then T is the identity tranformation on V. I'm ...
5
votes
3answers
147 views

Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution

Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution I want to prove it by Proof by Contradiction, but I am not sure how to proceed with the proof.
1
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3answers
44 views

Concise induction step for proving $\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$

I recently got a book on number theory and am working through some of the basic proofs. I was able to prove that $$\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$$ with the help of the identity ...
0
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1answer
53 views

Bijection between $\mathbb N \times \mathbb N$ and $\mathbb N$

Show that $\mathbb{N} \times \mathbb{N} \sim\mathbb{N}$. I found a bijection such that $g(k,l): \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ by $$g(k,l) = {(k+l)(k+l-1) \over 2} - (l-1)$$ But I am ...
8
votes
4answers
184 views

Proving $\int_0^1\frac{\log 2-\log\left({1+\sqrt{1-x^2}}\right)}{x}dx=\frac{\left(\pi^2-12\log^22\right)}{24}$

$$\int_0^1\frac{\log 2-\log\left({1+\sqrt{1-x^2}}\right)}{x}dx=\frac{\left(\pi^2-12\log^22\right)}{24}$$ At first, I think it can be calculated like the following one with differential method. ...
5
votes
1answer
146 views

Prove spatial velocity identity - screw theory

This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for ...
0
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0answers
18 views

Proving that two functions will not be equal after a certain point

I have two functions $f(p)=3-p\\ z(w)=q-w$ where $p \in \mathbb{N}$ and $1\leq p \leq 3$, $w \in \mathbb{N}$ and $1\leq w \leq 3$, $q \in \mathbb{N}$ and $q \geq1$ (none of them are constant) It ...
1
vote
3answers
49 views

If $f: A→B$ and $g: B→C$ are surjective, then $g\circ f$ is surjective.

In my homework, I wrote: Assume f and g are surjective. Let m be an element of C. then there exists a b that's an element of B, such that g(b) = m and an a element of A such that f(a) = b by ...
0
votes
1answer
51 views

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b)

I had this for homework before, and I wrote: There exists s,t, that are elements of integers, such that gcd(a,b) = sa + tb, so ngcd(a,b) = n(sa + tb) = s(na) + t(nb) but I got it wrong. How do I ...
1
vote
1answer
74 views

Prove that $\binom{p}{k}/p $ is integral for $k\in \{1,..,p-1\}$ with $p$ a prime number

I started by induction on $k$ For $k=1$ then : $1\in \mathbb{N}$ For $k=2$ then : $\frac{(p-1)}{2!} \in \mathbb{N}$ , indeed for all $p>{2}$, $p-1$ is even. (We still have $k<p$ it's ...
-2
votes
1answer
54 views

Proof by induction of this formula? [duplicate]

$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N}$ U ${0}$. I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction. I tested the base case where it's equal to zero, and it ...
0
votes
1answer
30 views

Area of a figure in terms of height and width

Attempting to give a thorough geometric answer to this question made me wonder: Can the area of any two-dimensional figure be expressed as a scaled product of its height and width? That is, for an ...
0
votes
2answers
23 views

Prove proposition on real numbers and inverses.

Prove the following proposition Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$ So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.
8
votes
1answer
209 views

question on translation of operator proof

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
59 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 ...
3
votes
0answers
107 views

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
104 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
44 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
1answer
12 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
53 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
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1answer
18 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
37 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
61 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
29 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
81 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 ...
1
vote
1answer
37 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
126 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
1answer
66 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 ...
3
votes
2answers
57 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
96 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
-2
votes
1answer
69 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
39 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
209 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
26 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 ...
2
votes
2answers
79 views

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

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
43 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
44 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
554 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 ...
2
votes
1answer
28 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
27 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
23 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 ...
-2
votes
1answer
34 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
24 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
17 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 ...