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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0
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0answers
10 views

A question about an special kind of proof [on hold]

Is there some proof method in which I can prove the existence of a mathematical object? How can I prove a conjecture that it is an existence statement of a mathematical object? Which are the ...
0
votes
1answer
742 views

Proof for Symmetry property of Congruent Segments

I am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises in ...
-1
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0answers
5 views

cone of influence and compact support

We are given that wavelet $\psi$ is of compact support $\subset$ [-k, k] then using the concept of cone of influenc how we consider its support to be $\{(x, y);|y-x|\leq |x|k\}$ .
0
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0answers
30 views

Differentiability and $L^1, L^2$ spaces

If $f\in L^1(\mathbb{R})$ then $\frac{d}{dx}\{f(x)\}\in L^1(\mathbb{R})$ where we have given that $f$ is of compact support.
9
votes
4answers
1k views

What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz ...
20
votes
1answer
255 views

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$. By plotting graph I have seen that there are no roots for $x$. Can somebody prove it theoretically?
18
votes
10answers
3k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
0
votes
1answer
20 views

Giving restriction on the value of $N$ in $\epsilon-N$ proof.

I'm still working on $\epsilon-N$ proof. If you don't mind is it possible for us to give restriction on the value of $N$ as illustrated by this example: Say after some manipulation of the limit ...
3
votes
1answer
726 views

How to prove if log is rational/irrational

I'm an English major, now doubling in computer science. The first course I'm taking is Discrete Mathematics for Computer Science, using the MIT 6.042 textbook. Within the first chapter of the book's ...
3
votes
1answer
30 views

General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
-1
votes
3answers
232 views

The limit of $\sin(1/x)$ as $x\to 0$ does not exists

Prove that the following limit does not exist. $$ \lim_{x\to 0} \sin\left(1 \over x\right) $$ Our definition of a limit: Let $L$ be a number and let ${\rm f}\left(x\right)$ be a function which is ...
0
votes
2answers
28 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
1
vote
1answer
31 views

Choosing the right N in $\epsilon-N$ proof

I'm just a little bit confused in choosing the right $N$ when working on the rough sketch of the proof. Suppose after some algebra we have reached the point where we get this expression, say: ...
3
votes
1answer
21 views

Partitioning a totally ordered set into three subsets according to the order

Consider a set $S$, and a total order $R$ over that set. Part (a) Given some element $e \in S$, explain why it is possible to partition $S$ into the following three sets: $$S_1 = \{ x \in S ...
4
votes
1answer
579 views

Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
3
votes
0answers
47 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
-1
votes
1answer
31 views

Set Theory Proof [on hold]

How can I prove that $A \cup B = A \cup (A'B)$ It is a useful identity when proving : $ P(A \cup B) = P(A)+P(B)-P(AB)$ But I don´t know where that identity comes from. Is there an analytical proof ...
0
votes
1answer
23 views

Let $a$ and $b$ be coprime positive integers. Prove that, for any integer $n$, there exist integers $s$ and $t$ such that $sa + tb = n$

I always sort of took this fact for (well..) fact. Can someone help me with the proof? Does this question have something to do with modulus? Since $a$ and $b$ are coprime ($gcd$ = 1), multiplying ...
-1
votes
0answers
43 views

Discrete Mathematics Proof about coefficients in a Function [on hold]

I need to prove these using any method. I know that it deals with the distribution of the coefficients of a function. I just don't exactly know about how to go about proving this. Let the function ...
0
votes
4answers
74 views

If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$

I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the ...
2
votes
1answer
62 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
1
vote
1answer
247 views

Finding the probability that X will be successful if its success is predicted

Consider an electronics company is planning to introduce a new camera phone. The company commissions a marketing report for each newproduct that predicts either the success or the failure of the ...
2
votes
1answer
163 views

Proving that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$ for all positive $n$

Hello I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction ...
0
votes
3answers
24 views

A property regarding intervals

While I was solving a problem on TopCoder I used the following assumption. I have n intervals: $ [a_1,b_1], [a_2,b_2],...,[a_n,b_n]$ and a number $T$ such that: $$ a_1 + a_2 + ... + a_n \leq T \leq ...
1
vote
1answer
44 views

Proof that ordinary multinomial coefficients rise monotonically to a maximum and then decrease monotonically

While most computations of ordinary multinomial coefficients for the following case require recursive summations, I found here a closed-form solution: $$(1+x+x^2+\cdots+x^q)^L = \sum_{a \geq 0} ...
0
votes
1answer
53 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
0
votes
0answers
36 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
4
votes
1answer
74 views

how to solve this elementary induction proof: $\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\le\ 2-\frac{1}{n}$

this is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction the question; $$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ ...
1
vote
0answers
7 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
1
vote
0answers
34 views

prove the Laguerre–Gaussian equation [closed]

How does one prove the Laguerre–Gaussian equation? $$ x\frac{d^2y}{dx^2}+\left(1-x\right)\frac{dy}{dx}+ny=0 $$
-2
votes
0answers
35 views

Show that something follow power law [closed]

I want to show that a and b follows power law. What do I have to do to show it? How people claim that something follows power law? Any direction would be a great help. The question is pretty ...
7
votes
4answers
172 views

Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$

Question:Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6\dots$, constructed by including the integer $k$ exactly $k$ times. Show that $a_n = ...
6
votes
1answer
74 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
19
votes
3answers
335 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
0
votes
1answer
68 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
1
vote
3answers
36 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
0
votes
2answers
26 views

Proof d-regular graph has an equal number of vertices in its bipartition

Let $G$ be a $d$-regular graph. Suppose that $G$ is bipartite with bipartition $(A,B)$. Prove that if $d>0$ then $|A| = |B|$. Also why is this statement false when $d=0.$ I'm not sure how to show ...
4
votes
1answer
68 views

Let R be a relation on set A. Prove that $R^2 \subseteq R <=>$ R is transitive $<=> R^i \subseteq R ,\forall i \geq 1$

this is my first question here. I'm still relatively new to more advanced mathematics and don't have much experience with proofs yet. I'm self-studying at the moment and therefore have no one to check ...
1
vote
2answers
26 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
2
votes
2answers
73 views

Better proof that $n \leq 2n$ for all natural numbers?

I tried proving via induction on naturals that $n \leq 2n$ for each natural $n$. Obviously, $0 \leq 2(0)$, and then assuming for any given $n$, $n \leq 2n$, you just show that $n + 1 \leq 2(n + 1).$ ...
0
votes
1answer
28 views

Prove $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism.

I'm working on proving the following claim: "Let $r \in U(n)$ and $\forall s \in \mathbb{Z_n}$, $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism." ...
1
vote
2answers
48 views

Discrete Mathmatics Proof

Here is the question: $a$ and $b$ are any two integers. $c$ is any prime. Prove that if $c$ divides $ab$, then $c$ divides $a$ or $c$ divides $b$ (or both, as in it can divide either or both, i.e. ...
0
votes
1answer
41 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
6
votes
3answers
159 views

Is there a simple way to illustrate that Fermat's Last Theorem is plausible?

A first step in proving a theorem is true could be to show that it is plausible, so at least you then would have a general idea that it could be true and have something to start with in proving it. ...
10
votes
7answers
3k views

What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi (A\cdot B)=\varphi (A)\cdot \varphi (B)$, if A and B are coprime? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its ...
2
votes
4answers
88 views

Proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$

So, as the title said I have to proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$. I have to do it with induction. So I got my start, for $n=0$: We have $1+4^0+7^0 = 1+1+1 = 3$ and ...
2
votes
2answers
34 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
2
votes
1answer
51 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
24
votes
6answers
1k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
0
votes
2answers
24 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...