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
26 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
3
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0answers
33 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
0
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0answers
12 views

How to prove unicity in a disjunction of $n$ propositions

Let's suppose I have the propositions $\varphi_1, \varphi_2,...,\varphi_n$ and I want to prove that there happens exactly one of them. How do you do it? To do it the hard way I guess we first need to ...
3
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2answers
148 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 ...
0
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0answers
13 views

Proof that every subset of extended real numbers has in set of extended real numbers supremum and infinum

I need to prove this statement, but I don't know how to prove it formally correctly. Can someone help how to prove it formally? We have linear ordering $(\mathbb R^{*},<)$, where $\mathbb R^{*} ...
0
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0answers
35 views

Refining Proof Methods [on hold]

Question: Can any problem be proven with a metamathematical proof or indirect proof rather than a direct proof? Must one call upon a "hat trick" theorem to beg his proof or for any problem can a ...
0
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2answers
23 views

How to show if a language is infinite, then there is no upper bound on the length of words in L?

L is a language over a finite alphabet. How to show that if L is infinite, then there is no upper bound on the length of the words within L? Can someone help me prove this.
0
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1answer
9 views

Deriving an expression for an n-th composition of some Mobius transformation

Let $T(z)=\dfrac{1-3z}{z-3}, T^2(z)=T(T(z)),..., T^{n+1}(z)=T(T^{n}(z)),n=1,2....$ Could anyone advise me on how to find an expression for $T^n(z) \ ?$ I'm trying to make use of the fact that there ...
2
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1answer
25 views

algebraic numbers and their squares

I'm trying to prove that if x is an algebraic number then x^2 must also be. It seems intuitive but I just can't find any kind of proof as I keep running into equations with fractional exponents that ...
0
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0answers
8 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...
0
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0answers
19 views

True or False: If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then $(g\circ f)''(a)=g'(f(a))f''(a)+g''(f(a))(f'(a))^2$

True of False: If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then $(g\circ f)''(a)=g'(f(a))f''(a)+g''(f(a))(f'(a))^2$. I wasn't sure if my interpretation of this problem was ...
2
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1answer
22 views

Proof verification for $fgh=1_A\dots\implies f,g,h$ are all bijections. - Cohn - Classic Algebra Page 15

Is the proof below correct? Thank you for your time! Notation: $xfgh\equiv h(g(f(x)))= (h \circ g \circ f)(x)$ Theorem: If $f:A\to B, g:B\to C, h:C\to A$ are three mappings such that $fgh=1_A$, ...
1
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2answers
19 views

Help with proof of the existance of a graph produced from deleting edges

Prove that every connected graph with an even number of vertices can be transformed into a graph with uniform degree 1 by only deleting edges. I have tested this with pen-and-paper and it seems to be ...
1
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2answers
32 views

Prove that if p divides xy then p divides x or p divides y

I am given that the following proposition is true. (Proved in class) "Suppose that $x$, $y\in \Bbb Z$, not both zero. Then there exists $m$, $n\in\Bbb Z$ such that $$mx + ny = d$$ where $d$ is the ...
-1
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2answers
29 views

prove cauchy-schwarz inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
2
votes
2answers
54 views

Prove by mathematical induction that exponentials grow faster than polynomials

How to prove that for $\forall q>1$ $\forall k\in \mathbb{N}$ $\exists c>0$ $\forall \in \mathbb{N}$ $q^n≥cn^k$? I should use mathematical induction.
2
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1answer
42 views

Can there be more than one proof for the limit as x approaches 3 of x squared equal 9?

Can there be more than one proof for this question? An answer has been provided here and I can see that proof is valid: ...
0
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2answers
19 views

Proof involving greatest common divisor [on hold]

Suppose that $\text{gcd}\:(a, y) = 1$ and $\text{gcd}\:(b, y) = d$. How do I show that $\text{gcd}\:(ab, y) = d$?
2
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0answers
20 views

Set $E$ which halves the measure of an open interval [duplicate]

This was an exam question. I know that my answer is wrong, but I believe myself to be on the right track. Can someone help me finish my construction? Here is the question. Find a set $E$ with the ...
1
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1answer
27 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
3
votes
1answer
43 views

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. Is there a general proof method to prove this ...
0
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1answer
24 views

Proving two graphs are isomorphic in polynomial time - Bondy/Murty - Graph Theory Page 6

I am trying to do the below problem: Now I can't see how one does this. I know you can explicitly show the bijections, but I can't see an easy way to do this, since it is $3\text{-regular}$. I ...
1
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1answer
25 views

Application of Riemann mapping theorem

Let $\Omega \neq \mathbb{C}, \emptyset$ be a simply connected domain and $a \in \Omega.$ Let $f:\Omega \to \mathbb{D}$ be a conformal map such that $f(a)=0, f'(a)>0.$ Could anyone advise me how to ...
0
votes
2answers
36 views

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ in some neighborhood of $x_0$

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ for all $x$ some neighborhood of $x_0$. My attempt is below. From the assumptions above, we have that $f(x_0) > M = f(x_1)$ for ...
1
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2answers
23 views

Even function divided by Odd function is an Odd function PROOF?

An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?
2
votes
3answers
37 views

Is this formula satisfiable?

I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not be necessary to write down a ...
7
votes
2answers
59 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
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0answers
8 views

Finding a Mobius transformation

Let $R=\{z\in \mathbb{C}: Re(z)>0, |z-3|>1\}$ and $A=\{z \in \mathbb{C}: 1<|z|<p\}.$ Find a Mobius transformation $f$ and $p$ such that $f$ maps $R$ conformally to $A.$ May I verify if my ...
0
votes
0answers
10 views

Showing that a collection of m solutions is linearly independant

Show that a collection $ \Phi_1 .. \Phi_m $ : I-->R of continuous functions satisfying $ \\ $ $ \int_I(\Phi_J(t)\Phi_k(t)dt $ =1 when j=k , 0 when j$\neq$k $ \\ $ is linearly independent. Multiply the ...
0
votes
1answer
15 views

Help needed to establish a conformal mapping

Could anyone advise me on how to find a conformal map from $H=\{z \in \mathbb{C}: Re(z)>0\}$ to $A= \{z \in \mathbb{C}:|z|>1, |z-2|<3\} \ ?$ I tried to compose the map in terms of ...
1
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2answers
38 views

Number of particles at time $t$

A following problem appears in my text book under the section of induction: At time $0$, a particle resides at the point $0$ on the real line. Within $1$ second, it divides into $2$ particles that ...
0
votes
3answers
46 views

List one of the ways in which Mario could buy the stars and comets. Note: Mario needs to spend all of his gold coins

Mario has 773500 gold coins to purchase a number of stars and comets. Each star costs 299 gold coins, and each comet costs 208 gold coins. If the number of stars that Mario buys is at least twice the ...
1
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2answers
49 views

Prove: $f(x)=e^{ax}$ is continuous on $\mathbb{R}$

Am I being fooled by how simple this statement looks? My book is currently telling me to take both $\lim_{x\rightarrow 0} f(x) =1$ and $f(x_1+x_2)=f(x_1)f(x_2)$ where $-\infty<x_1,x_2<\infty$, ...
0
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0answers
27 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
0
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2answers
40 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
1
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1answer
17 views

Proof verification for $f$ & $g$ surjective implies $fg$ surjective - Cohn - Classic Algebra Page 15

Question: Is this a valid proof? Side question: Am I less likely to get answers based on using notation $xfg=g(f(x))$? I want to prove that if $f$ and $g$ are surjective, then $fg$ is ...
0
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2answers
31 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
0
votes
1answer
34 views

Question on Proofs of Sets. [on hold]

The set $A$ is a subset of the set $B$ iff $A \cup B = B$ If $A$ is a subset of the set $B$, then $A \cup C$ is a subset of $B \cup C$.
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2answers
66 views

About the rationality of $1.1010010001\dots$ [duplicate]

Let's define $\rho=1.1010010001\dots$ which can be expressed by: ...
0
votes
4answers
80 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
0
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1answer
22 views

Proof verification: (Pretty pictures :) )Showing two graphs are not isomorphic. Bondy/Murty - Graph theory Page 5

I want to show the below two graphs are not isomorphic. Treat the left as graph $\bf G$ and the right as graph $\bf H$. $\bf G$ and $\bf H$ are not isomorhpic: Although we have a bijective mapping ...
0
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2answers
21 views

If $G$ is simple, then $\epsilon \leq {v \choose 2}$ - Bondy/Murty - Graph Theory with Applications Page 4

Question: Does this proof hold? Is this a bad proof? Any nicer proofs that don't rely on other theorems? Notation: $\epsilon$ - Number of edges $v$ - Number of vertices G - Here, any Graph ...
0
votes
1answer
17 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
1
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2answers
28 views

Induction on the number of marbles in a heap.

Here is the problem in full: "A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
3
votes
3answers
229 views

Show that the function is not continuous anywhere

I'm trying to prove that a specific function $f$ is not continuous for any $x_0$ that it is defined for. Here's what I have so far. Let $$f(x) = \left\{ \begin{array}{lr} -1 & : x\ ...
0
votes
1answer
37 views

Proving integration formulas from scratch

Prove the following integration formulas from scratch? (I uploaded them)
1
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3answers
52 views

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$ for all real values of $y$. I attempted to show this by finding the power series expansion of $\sqrt[4]{1+y^4} $ and then relating that to $1+|y|$; however, I ...
0
votes
1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
0
votes
0answers
17 views

A question about a change of variable

I have came across this question while trying to find the derivate of the inverse functioin. And I have found the following limit: $$ \lim_{y\to y_0} = \frac{1}{\frac{f(x) - f(x_0)}{x-x0}}$$ We also ...
3
votes
3answers
274 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...