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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1answer
61 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
1
vote
2answers
59 views

Can someone explain to me why set proof involve the words “or” and “and”

For example, on proving the distributive law of set theory, the following constitutes as a proof Proof : I am new to proof involving sets but this to me seems nothing more than replacing unions ...
-1
votes
1answer
28 views

Proofs of n^2 rem 4 [duplicate]

Show that if n is an integer than the remainder $(n^2 rem 4)$ = 1 or 0. I don't under what rem means in this form. Would it be n^2 + 4 = 1 or n^2 + 4 = 0?
2
votes
2answers
54 views

Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side ...
0
votes
2answers
23 views

Define a relation $D_n$ on $S$ by $xD_ny$ if and only if $x\mid y$. Determine if it's a poset.

Here is the question I am currently working on (screenshot): I'd appreciate some suggestions/guidance for part (a), proving that $D_n$ is a partial order. Reflexive: Let $x \in \mathbb{Z}$ ...
4
votes
1answer
35 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
2
votes
0answers
20 views

Applying rotation invariant linear operators to spherical harmonics

In the article "On boundary condition for multidimensional diffusion processes" A Venttsel says: I can't see how one can "prove that any other harmonic of order $n$ may be represented as a linear ...
2
votes
3answers
32 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
2
votes
3answers
63 views

What are the logical underpinnings of the epsilon- delta definiton of limits?

I'm having trouble getting my head around the epsilon-delta defintion of limits. I learned about conditional statements and I know that in order for a conditional to be true , one of the following ...
2
votes
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
41 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
119 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
56 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
50 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
75 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 ...
1
vote
1answer
20 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
78 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
117 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 ...
1
vote
3answers
59 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
49 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
24 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 = ...
1
vote
1answer
38 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
42 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
59 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
119 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
41 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
159 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
75 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
51 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
83 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
109 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
90 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 ...
2
votes
4answers
105 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
63 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
65 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
56 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
46 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
117 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
59 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
62 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
55 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
35 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
68 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
56 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 ...