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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2
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2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
-1
votes
2answers
39 views

Empty set Velleman's exercises

Doing an exercise from Velleman's 'How to prove it' I ended up thinking about exercise 2.3.8: Given that there are sets $ I=\{2,3\}, A_2=\{2,4\},A_3=\{3,6\},B_2=\{2,3\},B_3=\{3,4\}$. What is ...
2
votes
8answers
117 views

Prove by induction that $\frac{n^3}{3}+\frac{2n}{3}$ is an integer. [duplicate]

The question that I am working on is: Prove that $\dfrac{n^3}{3}+\dfrac{2n}{3} \in \mathbb Z \ \forall \ n \in \mathbb N$ The method that I think would be will work for this question is that I ...
0
votes
2answers
53 views

Induction question regarding Universe

I was given a question that looks like this. Prove that for each $2 \le n\in \mathbb N$, if $X_1,\ldots,X_n$ are subsets of some universe $U$, then the following is true: $$(X_1 \cup\cdots\cup ...
3
votes
2answers
47 views

How To Tackle Trigonometric Proofs involving $4$th and $6$th powers?

How do I prove that $\cos^4A - \sin^4A+1=2\cos^2A$ $\cos^6A + \sin^6A =1-3\sin^2A\cdot\cos^2A$ I was going through a very old and very rich book of Plane Trigonometry to build a nice foundation for ...
1
vote
3answers
68 views

Induction Proof using factorials

Recall that for $n \in N$, $n! = 1 \cdot 2 \cdots n$. Prove the following for each $n \in N$: $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$$ I ...
0
votes
4answers
73 views

Prove for each $n\in \mathbb{N}, 1^3 + 2^3 +\cdots+ n^3 = \frac{n^2(n+1)^2}{​4} ​ ​​​$ [duplicate]

So I was given a proof by induction question and here is my attempt $$1^3 + 2^3 + 3^3+\cdots+n^3= \frac{n^2(n+1)^2}{4}$$ $n=1$ $1=1$ Induction step: Assume statement is true for $n=k$, show true ...
1
vote
1answer
13 views

Prove that an underdetermined system of cannot have a unique solution(Is this proof correct?)

I know I misspelled underdetermine but is this proof correct? How can I improve it either way? Side Remark: Anyone who is down-voting please can you understand I new to this site and somewhat ...
2
votes
6answers
76 views

Prove that $xy+yz+zx \leq x^2+y^2+z^2$

Prove that $xy+yz+zx \leq x^2+y^2+z^2$ . Hint: Use $\frac{a+b}{2}\geq\sqrt{ab}$ First I tried using the hint by setting $a=x$ and $b=y+z$, however this results in the inequality: $$x^2+y^2+z^2 ...
2
votes
2answers
98 views

Understanding Spivak's alternative proof that $|a + b|\leq |a| + |b|$

For example, in Chapter 1 - Problem 14c Spivak asks the reader to come up with a different alternative proof that $$|a + b|\leq |a| + |b|$$ and this is what I found in the solution manual (with my ...
0
votes
3answers
57 views

proof by induction $2^n \leq 2^{n+1}-2^{n−1}-1$

My question is prove by induction for all $n\in\mathbb{N}$, $2^n \leq 2 ^{n+1}-2^{n−1}-1$ My proof $1+2+3+4+....+2^n \leq 2^{n+1}-2^{n−1}-1$ Assume $n=1$,$1 ≤ 2$ Induction step Assume statement ...
0
votes
4answers
44 views

A tautology that contains quantifier and logical connective.

It might seem a stupid "question", but I need a logical explanation of it. If $p(x)$ is a predicate and $q$ is a statement, then $(\forall x:p(x))\wedge q\iff \forall x:(p(x)\wedge q)$, and ...
0
votes
0answers
21 views

Is there any relationship between a worst matrix and its size and what are their common structures?

I am currently trying to test and calculate the worst possible $\mathcal{O}(f(n))$ for some algorithm. In order to do so, I need to find the worst possible (0,1) n x n matrix for some $n$s (e.g. ...
1
vote
0answers
12 views

boundary condition measure associated to a rotation invariant operator

According to A. Venttsel (On boundary condition for multidimensional diffusion processes) The measure in $(13)$ is of the form $\nu(drd\theta)\cdot d\varphi$ while in the general case we had ...
0
votes
0answers
10 views

Proof that $\partial_{t_1} F(1,w_0) = 0$ according to Venttsel

In the article "On boundary condition for multidimensional diffusion processes", A Venttsel says that I can't follow the author when he concludes that $a_1(1,w_0) = 0$. Do you have any ideas?
0
votes
0answers
27 views

Constructing an Algebraic Closure of a Field, $F$

I consulted a several sources on the internet, and they all begin with the construction of an extension, $F_1$ of $F$ such that each polynomial in $F[x]$ has a root in $F_1$. Specifically, $F_1 = ...
6
votes
2answers
41 views

Sujection, finite set, $|X| \le n$? [closed]

Suppose that $\{1, 2, \dots, n\} \to X$ is a surjection. How do I show that $X$ is a finite set and that $|X| \le n$?
1
vote
1answer
35 views

Define a relation — with functions and derivatives

Here is the problem I am working on: I am in a beginning level abstract math/proofs class, and haven't had much experience with calculus in any proof (or in any relation). Here is my understanding ...
0
votes
2answers
66 views

Define a relation $M$ on $\mathbb{Z} \times \mathbb{Z}$…

Update #2 (7.21.15): Here is a screenshot of the corrected question, in case anyone was interested. No need to look at the first update or original post to anyone viewing this for the first ...
1
vote
0answers
41 views

Subsets of emptyset

To follow up on an earlier question of mine and in order to improve understanding I would like to ask the following: What is the power set of $ \{\{\emptyset\}\} $? Is it $ \mathscr ...
0
votes
2answers
57 views

Can I argue that $g'$ is non zero in this case?

Consider two smooth maps $g,f$ given by $$ {\partial \over \partial x} g(x)= g'(x) = \int_0^1 {\partial \over \partial x} f'(u + t(x-u)) dt = \int_0^1 f''(u + t(x-u)) \cdot t dt $$ where $f' = ...
4
votes
1answer
102 views

Solving for $a$ in power tower equation

$$n=a^{(a+1)^{(a+2)^{(a+3)\cdots}}}$$ How would one go about solving in this equation? I am more used to solving equations in this form: $$n=a^{a^{a^{a\cdots}}}$$ Which you solve in this form: ...
1
vote
1answer
39 views

Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.

Here is my abstract maths problem. Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$. I am asked to either prove or disprove this statement. I am a little ...
1
vote
2answers
157 views

Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
3
votes
0answers
71 views

Help with proving that $\pi$ is irrational

I was trying to prove that $\pi$ is irrational, just to see if I could do it. So far, I've tried to do this by using the fact that the sum $$S=\sum\limits_{k=1}^\infty ...
0
votes
0answers
49 views

Prove FTC using limit of summation

It is not hard to show $$\int_a^bx^2\,dx=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n\left(a+(b-a)\frac kn\right)^2\right]=\frac{b^3}{3}-\frac{a^3}{3}.$$ With some effort one can also show ...
0
votes
3answers
80 views

Does $\Pr(\{X\leq x\})\geq\Pr(\{Y\leq x\})$ imply $\Pr(\{X\leq Y\})=1$?

Suppose that $(\Omega,\mathcal{F},P)$ is a probability space and $X,Y:\Omega\to\mathbb{R}$ are random variables satisfying $$ P(\{X\leq x\})\geq P(\{Y\leq x\}),\quad\forall x\in\mathbb{R}. $$ ...
0
votes
1answer
83 views

If AC and BC are two equal chords, BA is produced to P and CP cuts the circle at T the how is CT:CB=CA:CP?

I've been solving the following question, If AC and BC are two equal chords of a circle. BA is produced to any point P and CP, when joined cuts the circle at T then show that ...
-1
votes
4answers
77 views

For all integers $x$ and $y$, if $ x^3 + x = y^3 + y$ then $x = y$. [duplicate]

For all integers $x$ and $y$, if $x^3 + x = y^3 + y$ then $x = y$. This is what I have done so far: Proof: Suppose $x$ and $y$ are arbitrary integers. We know that $x^3 + x = y^3 + y$, we want to ...
1
vote
4answers
92 views

Proof - for all integers $y$, there is integer $x$ so that $x^3 + x = y$

For all integers $y$, there is an integer $x$ so that $$x^3 + x = y.$$ This is what I have done so far: Proof: Suppose $y$ is some integer. We want to prove that $$x^3 + x = y$$ for some integer ...
4
votes
2answers
61 views

Proof: $Y$ stochastically dominates $X$ implies $E[\phi(Y)]\geq E[\phi(X)]$ for increasing $\phi$

Suppose $X$ and $Y$ are real random variables with CDF $F$ and $G$ such that $F(x)\geq G(x)$ (i.e. $Y$ exhibits (first-order) stochastic dominance over $X$). Then, for all increasing function ...
0
votes
4answers
64 views

Understanding the proof technique of $A\cup (B\cap C)\subseteq (A\cup B)\cap (A\cup C)$.

I used to learn it in a different way; \begin{align} x\in A\cup (B\cap C)&\implies x\in A \textrm{ or } (x\in B \textrm{ and } x\in C)\tag{1}\\ &\implies (x\in A \textrm{ or } x\in B) ...
1
vote
0answers
15 views

Is there a specific and useful strategy for this kind of general setup of a problem?

I want some help in building a strategy to prove/disprove a statement in a specific problem. The specific problem setup is invisible to this question. In my problem, I have a sequence of smooth, ...
0
votes
1answer
49 views

How to prove the External Bisector Theorem by dropping perpendiculars from a triangle's vertices?

I've found two different methods to prove Internal Angle Bisector Theorem, viz. Wikipedia ("Proof 2") method and AskMath.com method. How can we prove External Angle Bisector Theorem with ...
1
vote
2answers
45 views

Simple proof by contradiction

I feel like I'm almost there, but I don't know what to right after this: for all real number $x$, if $x^2-2x\neq-1$, then $x\neq-1$. Let $p(x)$ be $x^2-2x\neq-1$ Let $q(x)$ be $x\neq-1$, My ...
-1
votes
3answers
115 views

$\varepsilon$-$\delta$ proof of limit (Spivak) [closed]

Could you please help me with this proof? $$ f(x)=x^4 + \frac1x $$ Prove using $\varepsilon$-$\delta$ that $$ \lim_{x \to 1} f(x)=2 $$ This problem is from Spivak's book Chapter 5 problem $3-iii.$ ...
0
votes
0answers
58 views

Prime divisors in Andy Loo's proof…

http://arxiv.org/pdf/1110.2377v1.pdf I have one more question related to that proof. Look at the definition of the symbol ${s \brace r}$ (page 4). Why if $\frac{3n}{4}<p\le \frac{4n}{5}$, then $p$ ...
2
votes
3answers
58 views

Problem in proof of Chinese remainder theorem, and applying it.

Please don't mark it as duplicate. First read the whole question. So Chinese Remainder Theorem states that,: Let $n_1,n_2,...,n_k$ be $k$ positive integers which are pairwise relatively prime. If ...
1
vote
1answer
35 views

One more question about the prime factors of binomial coefficient…

Could someone explain me one more thing: why if $p$ is greater than $4n/5$, but less or equal to $n$, then $p$ does not divide $\binom{4n}{3n}$? thank you in advance!
0
votes
1answer
48 views

How can you prove the triangle sum theorem?

Some things to consider: -This theorem has proved very, very many theorems many of which with trig, so you can't use any theorems that have been proven with the triangle sum theorem. -The Triangle ...
3
votes
1answer
32 views

Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...
3
votes
1answer
62 views

A few questions about Andy Loo's proof of existence of primes between 3n and 4n…

I have a few questions about Andy Loo's proof (get it here): why, for example, if $2n<p\le3n$, then $p$ does not divide $\binom{4n}{3n}$? Same situation for $\frac{4n}{3}<p\le\frac{3n}{2}$... ...
4
votes
2answers
47 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
1
vote
0answers
26 views

How to prove that sums of even powers is divisible by p

For $n\leq (p-2)$ I want to prove that $\sum_{k=0}^{p-1} (r+k)^{n} \equiv 0 \pmod{p}$ It is easy to see that it is true for odd n, since $(-a)^k \equiv -a^k$, and you can just pair up terms since ...
5
votes
1answer
227 views

Conjectured compositeness tests for $N=k\cdot 2^n \pm 1$ and $N=k\cdot 2^n \pm 3$

How to prove these conjectures ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ~\text{and}~ x ~\text{are ...
0
votes
2answers
84 views

Stuck on this proof that $ord(f) = ord(g)$

Let $f, g: \mathbb R \to \mathbb R$ be smooth maps such that $f(a) = g(a') = 0$ and let $\tau, \sigma : \mathbb R \to \mathbb R$ be diffeomorphisms such that $$ \tau \circ f = g \circ \sigma$$ ...
2
votes
3answers
67 views

How to prove $A=(A\setminus B)\cup (A\cap B)$ [duplicate]

How to prove $A=(A\setminus B)\cup (A\cap B)$. I have seen this problem and the solution is clear to me. Initially I was satisfied by my prove but now I think it is wrong. How I have proved ...
2
votes
6answers
1k views

How to prove this approximation for a logarithm? [closed]

I need to prove this approximation, but I am unable to conclude $$\log \left(1+\frac{1}{n}\right) \approx \frac{1}{n}$$
1
vote
0answers
51 views

Proving that the g.c.d of non-consectuive Fibonnaci numbers is also a Fibonacci number

I'm trying to prove that: for non-consecutive Fibonacci numbers, and I know that consecutive Fibonacci numbers are co prime, but I just don't how to prove this using what I know. **EDIT: Lulu has ...
7
votes
4answers
212 views

Looking for a direct proof of the following exercise

A friend of mine told me about the following problem: Let $\{r_n\}$ be a sequence of rational numbers such that $\lim_{n\to\infty}r_n=x\in\Bbb R,$ $r_n\neq x,$ for every $n\in\Bbb N$ and ...