Tagged Questions

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
22 views

Equivalence of definitions of torsion of a curve

The classical definition of torsion of a curve is $\tau(s)= -B´(s)\cdot N(s)$ where B is the binormal vector and N is the normal vector but I´ve seen another definition of torsion: $\tau=lim_{\Delta ...
1
vote
1answer
47 views

How to interpret the logic of an “or” in a matrix proof.

I am trying to learn to better interpret the meaning of equations and that is the purpose of this question, not just to find the proof, but to find the logical flow of the proof and understand it. I ...
3
votes
1answer
33 views

Suppose a $\in \mathbb{Z}$. $a^{2}|a$ if and only if $a \in \{-1,0,1\}$

Suppose a $\in \mathbb{Z}$. Then $a^{2}|a$ if and only if $a \in \{-1,0,1\}$ So, I have started and this is what I have so far: Case 1: If $a^{2}|a$, then $a \in \{-1,0,1\}$. For the sake of ...
1
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2answers
26 views

Next step in proof of sets

Proposition to prove : (A-B)∩(B-A) = 0 So, I understand why this is 0, I'm just not sure what propositions should be used in proving so. I have this so far 1)(A-B)∩(B-A) :Premise ...
2
votes
2answers
32 views

Supposed a,b $\in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.

Supposed a,b $\in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even. I'm kind of stuck on the best way to get this started. My thinking is that I could use cases. i.e. ...
0
votes
2answers
31 views

Prove that if a $\in \mathbb{Z}$ then $a^{3} \equiv a(mod 3)$

Prove that if a $\in \mathbb{Z}$ then $a^{3} \equiv a(mod 3)$ So, the ways I have learned (or am learning, rather) to do proofs is using direct, contrapositive and contradiction. So, I started it ...
1
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4answers
42 views

Given: $a_{1} =1 $ and $a_{n+1}=0.5(a_{n}+x/a_{n})$ How to prove that $\lim_{n \to \infty}a_{n}=\sqrt{x}$?

I have the sequence $a_{1} =1 $ and $a_{n+1}=0.5(a_{n}+x/a_{n})$ with $x \in \mathbb{R} $ and $x>0$. Then it is given that $a_{n}^2 \ge x$ for all $n\ge2$ and $a_{n+1}\le a_{n}$. How can I show ...
0
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0answers
14 views

3-Dimensional proof of Miquel's Theorem?

I was watching a topology lecture and the lecturer claims that it is possible to prove Miquel's Theorem as follows. Miquel's Theorem (statement): Consider any 3 mutually intersecting circles (where ...
0
votes
2answers
44 views

How to prove Kleene star to be uncounably infinite?

Hi I have a language $L = \{a, b\}$. How can I prove that the Kleene star (set of all words over the language) of this language is uncountably infinite or countably infinite?
0
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1answer
15 views

Building a non-hamiltonian graph of $p$ vertices of $\frac{p-1}2$ degree each.

I want to build some graph with $p$ vertices all with degree of atleast $\frac{p-1}{2}$ that isn't hamiltonian. I imagine this is possible, but I can't seem to do it. Any suggestions? Perhaps looking ...
0
votes
0answers
27 views

Proof for $\gcd(F_m,F_n)=F_{\gcd(n,m)}$ [duplicate]

I saw many questions/answers, where: $$\gcd(F_m,F_n)=F_{\gcd(n,m)}$$ is taken as a fact. But how can I actually prove that this is true?
0
votes
1answer
43 views

Show that $(A \oplus B) \oplus B = A$

I having some trouble proving $(A \oplus B)\oplus B = A$, I understand the truth table logic but can someone example to me in theory what the equation mean in set theory?
3
votes
1answer
52 views

Prove that $\lim_{x\to 2} \frac{1}{x} = \frac{1}{2}$

I want to prove this limit by using $(\epsilon,\delta)$ definition $$\lim_{x\to 2} \frac{1}{x} = \frac{1}{2}$$ Here is what I have done $$|\frac{1}{x} - \frac{1}{2}|<\epsilon \Leftrightarrow ...
1
vote
2answers
43 views

How to prove that $x^2+1 \ge 2x $ for $x>0$?

How to prove that $x^2+1 \ge 2x $ for $x>0$? It seems obvious to me but I don't know exactly how to prove it. Any help would be great.
0
votes
2answers
25 views

Proof by induction with variable other than $n$

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
1
vote
3answers
31 views

Prove $\lim_{n \rightarrow \infty} \frac{1}{n}\cdot \frac{3 + \frac{1}{n}}{4 - \frac{1}{n}} = 0 $

Prove $\lim_{n \rightarrow \infty} \frac{1}{n}\cdot \frac{3 + \frac{1}{n}}{4 - \frac{1}{n}} = 0 $ Let $\epsilon > 0$ be arbitrary. I want to find $N$ such that $n \in \mathbb{N}$ guarantees $ ...
0
votes
0answers
20 views

prove or disprove these statements [on hold]

how can we prove or disprove these two claims? ([x] is floor of x) 1) for all x in R, for all e in R+, there exists some d in R+, for all w in R, if |x-w| < d --> |[x] - [w]| < e 2)the negation ...
2
votes
1answer
48 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
2
votes
2answers
41 views

Show that $\int_{x=a}^{x=b} f'(x) g(x) dx=f(b)g(b)-f(a)g(a)-\int_{x=a}^{x=b} g'(x)f(x)\, dx$

I have to prove the following: Suppose $f$ and $g$ are differentiable on $[a,b]$ and $f'$ and $g'$ are integrable on $[a,b]$. Prove that $f'g$ and $g'f$ are integrable on $[a,b]$ and that of: $$ ...
1
vote
2answers
32 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 ...
5
votes
0answers
77 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: ...
1
vote
0answers
30 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 ...
4
votes
2answers
316 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
votes
0answers
36 views

Refining Proof Methods [closed]

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 ...
2
votes
2answers
54 views

If $n$ is composite then $n$ divides $(n-1)!$ [duplicate]

We need to prove that if $n$ is a composite number $>4$, then $n|(n-1)!$. I wanted to ask if my observation is correct or not. What I think is that the statement can be reduced to $n|(n-2)!$ ...
0
votes
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
votes
1answer
14 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
votes
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
votes
1answer
29 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
votes
1answer
27 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
vote
2answers
23 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
vote
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
votes
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
59 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
votes
1answer
43 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
votes
2answers
21 views

Proof involving greatest common divisor [closed]

Suppose that $\text{gcd}\:(a, y) = 1$ and $\text{gcd}\:(b, y) = d$. How do I show that $\text{gcd}\:(ab, y) = d$?
2
votes
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
vote
1answer
34 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
50 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
votes
1answer
25 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
vote
1answer
26 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
39 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
vote
2answers
24 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
43 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
60 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
votes
0answers
10 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
16 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
vote
2answers
39 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
48 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 ...