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
47 views

If $a$ divides $b$, then $a$ divides $3b^3-b^2+5b$.

Prove: Suppose $a$ and $b$ are integers. If $a\mid b$, then $a\mid3b^3-b^2+5b$. I think I have an idea of how to prove this, but I'm not entirely sure. I can prove that each individual term in ...
0
votes
0answers
58 views

Well-defined functions using mod $p$ equivalence classes

Prove that if $m, n$ are elements in the set $\mathbb{Z}$ and $m \equiv n \pmod p$, then $m^2 \equiv n^2 \pmod p$. Also, is the function $f: [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = ...
0
votes
1answer
102 views

Equivalence classes of multiples of 3

I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition: for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3. 1) List four ...
0
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1answer
40 views

natural deduction on proving a claim

I am working on this proof and wanted to know if I am using the ID natural deduction rule correctly. Can I just assume B and A based on that rule? ...
1
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2answers
45 views

finding numbers to make both sides equal

Call a triple-x number an integer $k$ such that $k=x(x+1)(x+2)$ where $x \in Z$. How many triple-x numbers are there between 0 and 100,000? I thought by doing $8!$ and $9!$ would work to see how ...
3
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2answers
111 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
0
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1answer
45 views

Equivalence Relation and functions question

Could someone please confirm if I understand this correctly? Here is the problem: define ~ on Z by m ~ n in case m^2 ~ n^2. 1) What, if anything, is wrong with the following "definition" of a ...
1
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1answer
22 views

Proof of unique solution to a minimization over two sequences

Given two non-strictly ascending sequences, prove that no rearrangement of terms in either sequences will produce a smaller $S$. $A=1,2,3,4,5...\\B=2,3,4,5,6...$ $S=\left|A_1 - B_1\right|+\ldots ...
2
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1answer
93 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
1
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2answers
254 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
0
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2answers
70 views

Can Cantor's theorem prove that N is uncountable (paradox)?

I am struggling a bit trying to understand Cantor's theorem about the reals being uncountable. How can you choose a real number that is different from all real numbers in an enumeration $S$? I ...
3
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0answers
68 views

Theorem cannot be proven directly

Can we ever prove a theorem cannot be proven directly (i.e. We must use contrapositive or prove by contradiction.)? Can we even rigorously defined whether a proof is direct or not? Example: I was ...
0
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1answer
35 views

Not unit-speed curve on a sphere

This question has already been asked: Curve on a sphere but is slightly different because I don´t have the hypothesis that $\alpha(t)$ is a unit-speed curve; anyway I wanted to do it by myself ...
0
votes
2answers
100 views

How to prove that the diameter of a graph is less than 2 given that the minimum degree of any vertex in G is greater than the number of vertices / 2

How do I prove the following statement. Let $G = (V,E)$ be a graph. Prove that if $δ(G) \ge \frac{|V|}2$, then $\operatorname{diam}(G) \le 2$ I believe $\delta$ is minimum degree of any vertex ...
2
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1answer
70 views

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them ...
-1
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1answer
108 views

Knowing People Proof [closed]

If I choose any four students among a class, at least one of the four knows all of the other three. Prove that there must be a student who knows everybody in the class.
1
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2answers
57 views

Prove this limit of $x^4 + 1/x$ formally

prove: $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} $ So, $lim_{x\to 1} \space \space \space x^4 + \frac{1}{x} = lim_{x\to 1} \space \space \space x^4 + lim_{x\to 1} \space \space \space ...
0
votes
1answer
32 views

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$ I want to do the $n - 1 \rightarrow n$ induction step. But I'm confused as to what my base case is. Usually if I want ...
1
vote
3answers
34 views

Prove that a sequence has a limit

Sequence $a_{n}$ satisfies $|a_{n}| \leq n$ for all $n \in \mathbb{N}$. Let sequence $b_{n} = \frac{a_{n} + 5}{n^{2} + a_{n}}$, prove that $b_{n}$ has a limit, and find it. I know that $b_{n}$ has a ...
1
vote
1answer
85 views

Manipulation of Bell Polynomials

Like lets say the following: $ g(n) = \sum_{k=1}^{n}B_{n,k}(f'(x),f''(x),.....,f^{n-k+1}(x)) $ What would $\frac{d}{dx} g(n)$ be? Or perhaps how to expand a Bell polynomial in this form into ...
0
votes
1answer
61 views

Prove that if $\{u_1, u_2, u_3\}$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ and we have $c_1u_1+ c_2u_2+c_3u_3 = 0$, then $c_i=0$. [duplicate]

Prove that if $\{u_1, u_2, u_3\}$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ and we have scalars $c_1, c_2, c_3$ such that $c_1u_1+ c_2u_2+c_3u_3 = 0$, then each of the scalars is equal ...
1
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1answer
29 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
55 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
41 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 ...
2
votes
2answers
33 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
68 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
44 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
vote
3answers
53 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
votes
0answers
39 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 ...
1
vote
2answers
91 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
19 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
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0answers
60 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
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1answer
49 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
66 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
1answer
47 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
39 views

Proof by induction with variable other than $n$ [duplicate]

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
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3answers
33 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 $ ...
3
votes
1answer
93 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
52 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
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2answers
53 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
votes
3answers
186 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
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0answers
40 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
571 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 ...
2
votes
2answers
67 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
30 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
20 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
26 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
45 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
34 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
30 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 ...