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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1
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2answers
35 views

Not all ideals are finitely generated

Let $R=\{a_0+a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, a_o \in \mathbb{Z}, n\in \mathbb{Z}_{\geq 0} \}$ and $I=\{a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, n\in \mathbb{Z}^{+} \}.$ ...
1
vote
1answer
136 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://www.math.uga.edu/~pete/2400calc2.pdf. This theorem can be ...
1
vote
3answers
125 views

Arithmetic and geometric mean

I need to prove that for $a=\frac{x+y}{2}$ and $g=\sqrt{xy}$, following statments are true or false: For $x\not =y,a>g$ and $x=y, a=g$. I have no idea how to do this, so any help is welcomed. ...
1
vote
1answer
149 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
1
vote
1answer
350 views

Combinatorially prove something

So i'm not sure at all how to prove things using a combinatorial proof. Where to do i start? What do i need to think about etc. For example how would i prove $$\sum_{i=0}^n {n \choose i} 2^i = 3^n ...
1
vote
1answer
409 views

Find a bijection from $(A^B)^C$ into $A^{B \times C}$ [duplicate]

Possible Duplicate: How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers? Notation: Let A and B be sets. The set of all functions $f:A \rightarrow B$ is denoted by $B^A$. Problem: ...
0
votes
1answer
36 views

Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function? [on hold]

Let $A=B=\mathbb N$. Relation $R$ is: $(a,b)\in R$ iff for some $q \in \mathbb Z$ we have $a=5q+b$ Given a relation, show that it's a function. To Show: $\forall a \in A \ \exists b \in B$ such ...
0
votes
3answers
32 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
votes
5answers
82 views

Series and sequences convergence with a certain condition.

Let $\sum_{n=1}^\infty (a_{n})$ converge. Let $\{n_{k}\}$ be a subsequence of the sequence of positive integers. For each $k$ define $b_{k}=a_{n_{k-1}+1}+...+a_{n_{k}}$ where $n_{0}=0$. Prove that ...
0
votes
1answer
41 views

Prove that there is a 1-1 correspondence between the set of subgroups of $\mathbb{Z}/N \mathbb{Z}$ and the set of the positive divisors of $N$

Im interested in the above Proof, is because I have the intiuition that it is not true at all, because for example, all the primes have exactly 2 positive divisors 1 an themselves, How Can I prove or ...
0
votes
1answer
33 views

Show that are logically equivalent

Can you answer me , please. 1- Show that (p→r)∧(q→r) and (p∨q)→r are logically equivalent 2- show that (p → r) ∨ (q → r) and (p ∧ q) → r are logically equivalent without truth table .
0
votes
1answer
42 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
0
votes
1answer
63 views

Proofing a Reachable Node Algorithm for Graphs

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
0
votes
3answers
118 views

How to prove if $n$ is prime and $n | a^2$ then $n | a$?

My professor assigned this for homework but I don't understand how to connect the dots. He suggested using the fact that $\gcd (x,y) \cdot \operatorname{lcm} (x,y) = xy$ but I'm not sure how that's ...
0
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1answer
50 views

Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
0
votes
4answers
74 views

Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$

Determine if $g$ is injective; surjective; bijective. Question on a recent test regarding one-to-one and onto functions. Was very difficult for me, could not even begin to answer either. This is ...
0
votes
0answers
54 views

Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
0
votes
1answer
57 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
0
votes
4answers
69 views

Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$

Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$ Extra question in my proofs homework, similar to another question I posted, but not exactly the same I don't think. ...
0
votes
2answers
48 views

Need help with a math proof

Any help would be greatly appreciated. Let $m,n,p,q \in \mathbb{Z}$. If $0 < m < n$ and $0 < p < q$ then $mp < nq$.
0
votes
2answers
101 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
votes
1answer
99 views

Natural deduction proof of a simple formula.

I am trying to prove the following formula using only natural deduction system: $$\vdash (A \supset (A \supset B)) \supset (A \supset B),$$ and this, according to natural deduction rules leads to $$A ...
0
votes
3answers
101 views

Math real numbers analysis.

Given a real number $x$, and a natural number $N\gt 1$. Consider the numbers : $0, x−\lfloor x\rfloor,2x−\lfloor2x\rfloor,\ldots,Nx−\lfloor Nx\rfloor.$ Show that some pair of these numbers differs ...
0
votes
2answers
164 views

Proof for $\epsilon$ closeness to $\sqrt{2}$ for all values of $\epsilon$

I am trying to understand the proof above, Proposition 7, given by the author, but I got stuck at some points. My explanation: Now, suppose for contradiction there is an $\epsilon > ...
0
votes
1answer
150 views

how to prove this scheduling problem

I need some hints for proving the correctness/optimality of the below homework problem. It is a task-schedulding problem with deadlines and penalties. There are n tasks, each of which has a deadline ...
-1
votes
3answers
591 views

How many ways can $5$ rings be placed on $4$ fingers?

I've been trying to solve this problem and I am kind of struggling with it and with other combinatorics problems. Could you check and see if i did it right? Given problem: How many ways can 5 ...
-2
votes
1answer
797 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
-3
votes
2answers
200 views

Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
29
votes
6answers
2k 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 ...
42
votes
7answers
2k views

Could I be using proof by contradiction too much?

Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it ...
11
votes
3answers
820 views

A good book for learning mathematical trickery

I've seen several question here on what book to read to learn writing and reading proofs. This question is not about that. I've been doing that for a while, and I'm quite comfortable with proofs. I am ...
56
votes
15answers
8k views

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that ...
31
votes
1answer
2k views

A proof of the Isoperimetric Inequality - how does it work?

Here is a nice proof of the isoperimetric inequality (equality part ommited): Isoperimetric Inequality If $\gamma$ is any simple closed piecewise $C^1$ curve of length $l$, with it's interior having ...
26
votes
12answers
4k views

Prove that a counterexample exists without knowing one

I strive to find a statement $S(n)$ with $n \in N$ that can be proven to be not generally true despite the fact that noone knows a counterexample, i.e. it holds true for all $n$ ever tested so far. ...
7
votes
6answers
3k views

Basic book about mathematical proofs

I'm looking for an understandable book about mathematical proofs. As an engineer/computer scientist major I'm used to do higher math on a daily base, but whenever I'm asked in an exercise to "show" ...
15
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16answers
1k views

Beautiful, simple proofs worthy of writing on this beautiful glass door [closed]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
14
votes
9answers
5k views

How Can One Prove $\cos(\pi/7) + \cos(3 \pi/7) + \cos(5 \pi/7) = 1/2$

Reference: http://xkcd.com/1047/ We tried various different trigonometric identities. Still no luck. Geometric interpretation would be also welcome. EDIT: Very good answers, I'm clearly impressed. ...
10
votes
4answers
410 views

Application of computers in higher mathematics

Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc. It is at all possible to delegate the thinking of a ...
5
votes
2answers
410 views

Solve equation $ 1+2^x=3^y$

Find integers $x$ and $y$ such that$$ 1+2^x=3^y.$$ It is obvious that $x = y = 1$ and $x = 3, y = 2$ are solutions. I think others are not. How to show that?
5
votes
6answers
9k views

Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + … + x y^{n-2} + y^{n-1})$

In Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've gone ...
27
votes
9answers
4k views

Rational + irrational = always irrational?

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + ...
7
votes
4answers
493 views

Book Recommendations and Proofs for a First Course in Real Analysis

I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is: How do we come out with a proof? Do we use some intuitive idea first and ...
4
votes
3answers
323 views

Proof that $\ln(n^2)(\ln(n) - 1) < n$ for all $n\in\mathbb{N}$

I would like to know which proof strategy to use when proving the next inequality: $\ln(n^2)(\ln(n) - 1) < n,\quad\forall n \in \mathbb{N}$. I have been trying to use this two proved inequalities ...
5
votes
2answers
335 views

How do you get a paper to be peer reviewed

I have a proof that I want to undergo peer review. I unfortunately am not affiliated with any university. How should I go about getting it reviewed and either rejected or published? Thanks!
4
votes
2answers
336 views

Proving that if one person in any group of four knows three, then someone knows everyone.

title can't exactly capture the description of this problem so well. Here's the question in full: "At a convention, any group of four people contains one who knows the other three. Prove there is ...
4
votes
2answers
829 views

Learning Proofs (for Computer Science)

Harvard's math curriculum, for freshmen, is divided into 4 classes beyond the BC Calculus level, Math 21, 23, 25 and 55. Math 21 is your classic plug-and-chug multivariable calculus and linear algebra ...
4
votes
1answer
1k views

Bounds of Euler's totient function?

Conjecture : Let $\phi(m)$ be Euler's totient function $1 \leq \phi(m) \leq \lceil \frac{m-1}{2} \rceil ~~$ if $~~m~~$ is even $\lceil \frac{m+1}{3} \rceil \leq\phi(m) \leq m-1 ~~$ ...
3
votes
0answers
52 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
19
votes
10answers
2k views

How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?

How would I prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
8
votes
2answers
279 views

An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...