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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1answer
64 views

Conditional proof/contradiction, long example problem

Here are the premises/conclusion, and where I've gotten so far. $1.$ $(W\wedge E)\rightarrow (P\vee L)$ (PR) $2.$ $(W\wedge \neg E)\wedge R))\rightarrow (P\vee D)$ (PR) $3.$ $((W\wedge \neg ...
1
vote
1answer
39 views

Asymptotic Normality of MLE when data is modelled with covariates

Say I have data vector $X_1,\ldots,X_n$ which I want to model with some parametric distribution function $f(X_i;\theta,Z_i)$ and covariates $Z_i$. In this case, how can I prove the asymptotic ...
0
votes
1answer
50 views

Real Analysis using the Archimedean Principle

Let $x$ be a real number. Then there exists a natural number $n$ such that $3^n > x$. proof: Let $x$ be a real number. Suppose that $x$ is an upperbound of the natural numbers. We know $1$ is a ...
0
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2answers
32 views

How do I begin to prove the limit of this definite integral?

I was fooling around with a graphing calculator, and I noticed a pattern in the functions of $$x^a+y^a=1$$ where $a$ is an even number. As $a$ increases, the graph begins to look like a square (if you ...
2
votes
3answers
48 views

Show that $G_{s}$ is a normal subgroup of $G$

Definition: $G_{s}:=\{g \in G: g.s=s\}$ My attempt is the following: We take $g \in G$, and we consider this two sets: $$gG_{s}:=\{gh:h\in G_{s} \}$$ $$G_{s}g:=\{hg :h\in G_{s}\}$$ and we will ...
4
votes
2answers
61 views

How to prove the following bounds expression

Let n be a positive integer. Prove that there are 2^(n−1) ways to write n as a sum of positive integers, where the order of the sum matters. For example, there are 8 ways to write 4 as the sum of ...
0
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1answer
23 views

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Determine all values of $T(9i) $

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Could anyone advise me how to find all possible values of $T(9i) \ ?$ A mobius ...
0
votes
1answer
22 views

Prove that a $\kappa : G/G_{s} \to G.s$ is a bijection

I have to prove that given an action this function $\kappa : G/G_{s} \to G.s$ is a bijection. $$ G/G_{s} \to G.s$$ $$gG_{s} \to g.s$$ Where $G$ is a group and: $G_{s}:=\{g \in G : g.s=s\}$(Isotropy ...
4
votes
1answer
74 views

Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.

Suppose $f:\Bbb R\to\Bbb R$ is continuous such that $f(x)\to0$ as $|x|\to\infty$. Prove that $f$ attains either a maximum or a minimum. My attempt at the question : Given $\epsilon > 0 \ \ ...
-1
votes
2answers
53 views

least value of a complex number

If $z_1,z_2,z_3,z_4\in C $ satisfy $z_1+z_2+z_3+z_4=0$ and $|z_1|^2+|z_2|^2+|z_3|^2+|z_4|^2=1$ then what will be the least value of $|z_1-z_2|^2+|z_1-z_4|^2+|z_2-z_3|^2+|z_3-z_4|^2$? What approach ...
3
votes
1answer
55 views

Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:Im(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$

Could anyone advise me on this problem: Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:\text{Im}(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$ ...
1
vote
2answers
109 views

Proof Question using Proof By Contradiction, irrationality of $a + \sqrt[b]{5}$

I answered this question but am not quite sure if i did what was correct. If anything is wrong please point it out, thanks. Question:Prove that any number of the form $a + \sqrt[b]5$ is irrational, ...
1
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3answers
89 views

Confirm definite integral equals zero $\frac{\sin(x)}{(1-a\cos(x))^{2}}$

Is this statement about the definite integral of a particular function $F$ true? $$\int_0^{2\pi}F(x)\, \mathrm{d}x = \int_0^{2\pi}\frac{\sin(x)}{(1-a\cos(x))^2}\, \mathrm{d}x = 0 \ \text{ for }\ ...
0
votes
2answers
50 views

Let A be an invertible nxn matrix. Prove that $\det(\operatorname{adj}(A^{-1})) = (\det(A))^{1-n}$

Let $A$ be an invertible $n\times n$ matrix. Prove that $\det(\operatorname{adj}(A^{-1})) = (\det(A))^{1-n}$ I tried starting with $A^{-1} = 1/\det(A) \cdot \operatorname{adj}(A)$ I tried everything ...
1
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0answers
74 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
0
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2answers
61 views

How to prove the following expression

Prove that if it takes you 5 minutes to solve any Sudoku puzzle and 14 minutes to solve a word search, you can completely occupy yourself on any flight of 52 minutes or longer provided that you have a ...
1
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0answers
57 views

suggestion on how to learn maths effectively

this question is not about a problem.My problem is I was made to read topics such as real analysis,complex analysis,metric spaces,topology,functional analysis,abstract algebra comprising of group ...
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votes
2answers
34 views

Use division algorithm and then induction to show 3|(n³+2n) for all ℕ. [duplicate]

For division algorithm, would I do something along the lines of n³+2n = 3q+r and go from there? For induction, I did the base case, which is true, and so then I moved on to the k+1 case, in which I ...
1
vote
1answer
75 views

Prove that if $p\ge 5$ is prime, then $p^2 + 1$ is composite

So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$? ...
1
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2answers
134 views

Prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$

I have to prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$. My attempt: We consider the function $g(x)=x^{3}-3x+b$.Now since it is a polynomial it is ...
2
votes
1answer
63 views

Prove that $p \ge 5$ is prime, then the remainder of $p$ upon division by $6$ is $1$ or $5$.

An example in my textbook, but I'm not quite sure how to set this one up, because of the $p \ge 5$ part. How do I start it off?
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votes
2answers
47 views

Is there an easy way to check convergence of real sequence? [closed]

How to check convergence of a sequence (for example, of $x_n =[nx]/n$ for a fixed $x \in \mathbb R$.) For series we can use various tests to check its convergence which are not available for ...
1
vote
0answers
20 views

Finding the highest power of N to fit in a given power of 2

I am trying to find the highest power $p$ of a number $N$ that will fit in a given power of 2. To give this some context, I am trying to find the largest power of $N$ that will fit in a 64-bit signed ...
1
vote
2answers
60 views

How to prove that if m and n are natural numbers than m+n is also a natural number?

Problem sounds easy enough - prove that if $m$ is in set of all natural numbers (let's call it $\mathbb N$) and so is $n$ than $m+n$ also must be there. Probably it should be done using induction. But ...
2
votes
3answers
95 views

Prove that $2\sqrt{n}\sqrt{n+1} < 2n + 1$ for all positive integers.

I've been testing this with many values and it seems to always be true. I've been trying to rework the inequality into a form where it's much more obvious that the left hand side is always less than ...
3
votes
3answers
103 views

Rational number arbitrarily close to a square root of 2 [duplicate]

I am trying to prove the proposition by contradiction For all rational $c > 0$, there exists a rational number $x$ such that $x^2 < 2 < (x + c)^2$. with the negation $x^2 \ge 2 \lor 2 ...
0
votes
1answer
78 views

Proof involving the Pigeonhole principle

Prove that among any given $n + 1$ positive integers, there are always two whose difference is divisible by $n$ My Answer: Using Pigeonhole principle: From a set of at least $2$ different $n+1$ ...
2
votes
1answer
60 views

How to prove uniqueness of *wannabe* final object in a slice category?

I am beginning to study category theory, and I think I need your help to find my way in this sea of uncertainty (!). I have the following problem (n. $5.11$ from Aluffi's Algebra: Chapter $0$). Let ...
0
votes
4answers
73 views

A intersection (A union B)

I'm trying to prove $A \cap (A \cup B) = A$. I'm stuck on the last part of my proof, not sure how to show next: $$x \in A \cap (A \cup B)$$ $$\iff x \in A \;\;\text{and}\;\; x \in A \cup B$$ $$\iff x ...
1
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3answers
105 views

Proof verification: A $(16,5,8)$ binary code does exist.

Well I have used spheres in coding with radius, $r=\left\lfloor\frac{\delta -1}{2} \right\rfloor=\left\lfloor\frac{8 -1}{2} \right\rfloor = 3$ and that means we have $\sum \limits_{i=0}^3 {16 \choose ...
0
votes
2answers
28 views

Is there some trick to manipulating an equation? (adding 0s, multiplying by 1, etc..)

I have such a hard time doing this sort of thing that it's annoying me. I'm not very mathematically inclined but it frustrates me that a solution with such a small answer takes me more than a page to ...
4
votes
3answers
118 views

Prove the gcd $(4a + b, a + 2b) $ is equal to $1$ or $7$.

So in the question it says to let $a$ and $b$ be nonzero integers such that $\gcd(a,b) = 1$. So based on that I know that $a$ and $b$ are relatively prime and that question is basically asking if the ...
2
votes
3answers
119 views

Counting the number of different ways in which groups of one or two can be formed…

I'm having trouble proving that the number of ways n>3 people can be divided into groups of either one or two is equal to: $A_n = A_{n-1} + (n-1)⋅A_{n-2} $ I'm trying to prove this by counting but ...
1
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3answers
74 views

Proof that a sequence is convergent

I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I proved that it is increasing through ...
1
vote
1answer
148 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
3
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1answer
54 views

Prove $\forall n \in N$, every set of natural numbers of size n has a maximum element. May assume that sets do not repeat numbers.

Prove using induction. So i'm a bit confused about how to do this question. My attempt at it seems like i'm missing a lot and it looked to easy. ...
1
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1answer
29 views

For integer $n$ prove that if there is no integer $m\le \sqrt{n}$ such that $ m | n$, then $n$ is prime.

At first I thought that the best way to prove this statement is to take the direct approach and show the subset {1, 2, 3,...sqrt(n)} and the subset {sqrt(n),... n/3, ..., n/2,...,n} and show that ...
2
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2answers
40 views

Proving that $\bigcup\mathcal{A}\times\bigcap\mathcal{B}\subseteq\bigcup\{a\times b\mid a\in\mathcal{A}\land b\in\mathcal{B}\}$ is always true

The problem is to prove that the following expression is true for any families of sets $\mathcal{A}$ and $\mathcal{B}$ that are not empty. ...
0
votes
2answers
64 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
0
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1answer
21 views

Having trouble with showing this CANNOT be a theorem in incidence geometry.

Consider the following statement: If l and m are any two distinct lines, then there exists a point P that does not lie on either l or m. (a) Show that this cannot be a theorem in incidence geometry. ...
6
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5answers
157 views

Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$ [duplicate]

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
5
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3answers
77 views

About using Archimedean property to prove the existence of least upper bound

I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property. Archimedean property. Let $x$ and $\epsilon$ be any positive ...
4
votes
1answer
87 views

Prove that the kernel is of dimension 2

"Experimentally", I found that the kernel (null space) of the following matrix is of dimension 2. I'd like to prove it, but haven't managed yet: \begin{equation} \text{for almost all } t>0,\quad ...
0
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2answers
16 views

Prove If LCM(a,b) = c and a|k and b|k then c|k.

Prove If LCM(a,b) = c and a|k and b|k then c|k. I know that c divides a and b if c = Least Common Multiple of a and b. I also know that c divides all multiples of a and b. I just am not sure how ...
0
votes
2answers
31 views

Prove this sum of binomial terms using induction.

Here's the problem stumping me today: Let $n \in \mathbb{N}$ and $r \in \mathbb{N}$ such that $r \leq n$, and prove using induction that $\binom{n+1}{r+1} = \sum\limits_{i=r}^n \binom{i}{r}$. I've ...
0
votes
2answers
39 views

Suppose that the $\lim a_{n} = 1$ and $x < y$

Suppose that $\lim a_{n} = 1$ and $x<y$. Is it possible to show using the Algebriac limit theorem that if the $\lim xa_{n} < y$. Then $0<x<y$? .
1
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0answers
39 views

no. of regions a plane is divided into by $n$ lines in general position

My notes state the Counting process for knowing no. of regions a plane is divided into by $n$ lines in general position := Let $h_1(n)=$ No. of parts a line is divided by $n$ distinct ...
4
votes
1answer
41 views

Prove [(xy=0)∧(x,y∈ℤ)]→[(x=0)∨(y=0)]

I'm working through a higher algebra textbook. It has some exercises related to the positive integers and I'm stuck on this proof. Here's what I have so far: Attempted proof Assume the contrary, ...
1
vote
1answer
16 views

$[x]_0,[x]_1\ldots [x]_n$ is a basis for vector space $V$.

here is a lemma which requires the use of falling factorials which are written as $[x]_n=x(x-1)\ldots(x-(n-1))$ : Lemma:Let $V$ be a vector space of polynomials over $\mathbb C$ , then ...
0
votes
1answer
47 views

show that if det()=0 with A, and B, then AB = BA.

show that if |b c| |a b| = 0 with A = [a a] [b b] , and B = [b b] [c c], then AB = BA. I'm not exactly sure how to go about proving this. I tried computing both AB and BA, then factoring ...