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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2answers
54 views

Prove that if $a<1/a<b<1/b$ then $a<-1$

The following is Exercise 3.2.8 from Velleman: Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<1/a<b<1/b$ then $a<-1$. I solved it using the hint in the back of ...
2
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0answers
25 views

$x-y^4= LCM(x, y)$ [duplicate]

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with the least common multiple, but other than that, the textbook gave no hints ...
0
votes
1answer
44 views

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group.

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group. To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition ...
1
vote
2answers
64 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
1
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2answers
58 views

Why is $f(x) = x^2$ uniformly continuous on [0,1] but not $\mathbb{R}$

According to How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity? There is a lot of agreement that $x^2$ is not uniformly continuous. But is $x^2$ uniformly ...
0
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1answer
21 views

Deriving the sum to product formula for sine using this method

I am trying to derive $sinC-sinD$ By this method: So far I have tried to set up the same method by beginning with $sin(A+B)-sin(A-B)$, but this reduces to a trivial zero and I can't find another ...
0
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2answers
59 views

Velleman's exercise $3.1.7$

Prove that if $a^3>a$ then $a^5>a$. Velleman gives this "hint": $$\text{One approach is to start by completing the following equation:}\ (a^5-a)=(a^3-a) \cdot x$$ I don't understand this ...
1
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4answers
85 views

Proving that the sum of the first $2n$ terms of the series $1^2 - 3^2 + 5^2 - \cdots$ is $-8n^2$ by induction

Use mathematical induction to prove the following for the first $2n$ terms of the series $$1^2 - 3^2 + 5^2 - 7^2 + \cdots = -8n^2.$$ As we have odd numbers that are squared we could use $n = ...
1
vote
1answer
44 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
3
votes
1answer
27 views

Show that $T$ is the Set of All Sets Using the ZF Axioms

Let x be a set. Define the "set" $S = \left\{ y:x\subseteq y \right\}$ and $T = \cup\left\{y:y\in S \right\}$. Given any set $w$, let $z=x \cup \left\{w\right\}$. Then $x \subseteq z$, so $z \in S$. ...
2
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1answer
59 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
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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
49 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
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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
34 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
31 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
62 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
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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
54 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
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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
75 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
16 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
101 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
58 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
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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
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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
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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
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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
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1answer
36 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
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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
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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
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2answers
58 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
103 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
81 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
90 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
78 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
94 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
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) ...