If you already have a proof for some result, but want to ask for a different proof (using different methods).

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Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [on hold]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
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0answers
34 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
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2answers
38 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
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1answer
35 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
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2answers
46 views

$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials

Prove that there exists constant $C>0$ that for all $f \in P_n$ we have: $$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$$ Where $P_n$ is space of polynomials with ...
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0answers
7 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
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3answers
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valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
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1answer
18 views

Looking for a way to improve my inductive proof of a statement derived by Rolle's Theorem

The following problem is 'absolutely' clear: Problem: Let $f$ be continuous on the interval $[a,b]$ and $n$-times differentiable on $(a,b)$ and $f$ vanishes on $n+1$ points $x_0< x_1 < \dots ...
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1answer
43 views

Accounts of the proof of Fermat's Last Theorem

I would like to collect a set of references to pieces of Wiles' 1995 proof of Fermat's Last Theorem. Has anyone recompiled the proof into another paper? Are there any books or articles that describe ...
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1answer
109 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
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3answers
62 views

Prove $-1$ and $1$ are the only units in $\mathbb{Z}$ [closed]

Prove $\mathbb Z^*=\{-1,1\}.$ I have a proof, which is posted as an answer below. I'm looking for an alternate proof.
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1answer
38 views

An alternative proof of the Tietze Extension Theorem(s)

Last summer I was working through a lot of Topology. I made it through the sections of my notes that dealt with separation properties, covering properties and continuous functions between spaces ...
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2answers
34 views

What is $\gcd(x,x+2)$?

Show that $\gcd(x,x+2)$ is $1$ if $x$ is odd and $2$ if $x$ is even. I am looking for a much simpler proof beside the one which I have posted.
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0answers
34 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
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0answers
38 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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1answer
33 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
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2answers
75 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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1answer
37 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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2answers
96 views

An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$

How does one prove the given series? $$\sum_{n=0}^\infty\left(\frac{(-1)^n}{4n-a+2}+\frac{(-1)^n}{4n+a+2}\right)=\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$$ This series came up in xpaul's ...
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1answer
39 views

Proof of easy matching condition for Hall's theorem

I was studying with the recitations provided in the course 6.042 "Mathematics for Computer Science" of MIT OCW and while studying the proof of Hall's marriage problem, I understood the first proof ...
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1answer
107 views

Proof of Faà di Bruno's formula using a convolution identity for Bell polynomials?

I have noticed there is an identity for Bell polynomials that can apply of Faà di Bruno's formula. This is a convolution identity that states: $$ (x \ast y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j ...
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9answers
342 views

Lesser known derivations of well-known formulas and theorems

What are some lesser known derivations of well-known formulas and theorems? I ask because I recently found a new way to derive the quadratic formula which didn't involve completing the square as is ...
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0answers
13 views

Linear independency question in $\mathbb{R}^3$ and cross product

If $u$ and $v$ are non-parallel vectors $\in \mathbb R^3$, is it true that the vectors $(u+v,u-v,u\times v)$ form a basis of $\mathbb{R}^3$? My answer would be affirmative supposing that $u$ ...
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4answers
48 views

Alternative proof for $\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$

Can I use only strong induction in order to prove $$\forall n\in\mathbb N (n\ge 6 \Rightarrow \exists a,b\in\mathbb N : n=3a+4b)$$ Is there any other option?
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1answer
29 views

Use division algorithm to prove for any odd integer n, $n^2 -1$ is a multiple of 8.

Here is what I know if n is any odd integer then $n$ can be expressed as $n=2k+1 ~~~ where~k\in\mathbb{Z}$.So $n^2-1=(2k+1)^2 -1=4k^2+4k=4k(k+1)$ but $k(k+1)~~ is~~even$. Thus $k(k+1)=2t, t\in ...
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1answer
47 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
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0answers
20 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
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1answer
71 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
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1answer
41 views

Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
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1answer
37 views

Need alternative proof to $ \exists x (k(x) \rightarrow t)$ entails $\forall x (k(x)) \rightarrow t $

I tried to prove $ \exists x (k(x) \rightarrow t)$ entails $\forall x (k(x)) \rightarrow t $ as; $ \exists x (k(x) \rightarrow t)$ $ \exists x (\neg k(x) \lor t)$ $ \exists x (\neg k(x)) \lor ...
0
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1answer
32 views

Elements of order 5 in $A_6$

I am trying to find the elements of order 5 in $A_6$ and I understand that they are of the form $(abcde)$, correct? So the number of elements is $(6*5*4*3*2)/5$=144. I looked somewhere else and it ...
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1answer
47 views

Placing bricks on Board

Situation: I have a $8\times 8$ board (sic), but two squares from it's one diagonal are removed (Black colored squares are removed) I'm given with plenty of(Rectangular) bricks having dimensions ...
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2answers
38 views

Prove that every nonempty finite set has a maximum.

how do I prove that every nonempty finite set has a maximum. I know how to explain this by words but didn't know how to put it into mathematical form. I found a way to prove this by induction in ...
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3answers
88 views

$\frac{1}{1+2}+\frac{1}{1+2+3}+\dots+\frac{1}{1+2+3+\dots+x}=\frac{2011}{2013}$

I want to see OTHER approaches than this one. Make sure they are significantly different and not a direct restatement. ...
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1answer
33 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
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2answers
46 views

Metrics and the Kuratowski closure axioms

Edit: Succinct proofs from user87690 can be found below, but I will gladly up-vote other valid approaches to any of the problems here! The following questions concern closure operators and the ...
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2answers
32 views

Proving $P(A|(B \cap C)) = P(B | (A \cap C)) P(A | C) / P(B | C)$ using Bayes' theorem.

The following equation can be proven rather uglily, provided that $P(B \cap C)$, $P(A \cap C)$ and $P(C)$ are non-zero, by expanding the conditional probabilities. $$P(A | (B \cap C)) = \frac{P(B | ...
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2answers
38 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 ...
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2answers
26 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 ...
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125 views

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
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3answers
45 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
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1answer
43 views

Set of open intervals in R with rational endpoints is a basis for standard topology on R

Show that the set $\mathcal{B} = \{(a,b) \subset \mathbb{R}: a,b \in \mathbb{Q}\}$ is a basis for the standard topology on $\mathbb{R}$ First I'll show that $\mathcal{B}$ is a basis on ...
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0answers
64 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
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0answers
121 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
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1answer
35 views

Set of all subsets of X that contain a set Q is a topology

Let $X$ be a set such that $Q \subset X$. Show that $\tau = \{\emptyset\} \cup \{U \in \mathcal{P}(X): Q \subset U\}$ is a topology on X. $\emptyset \in \tau$ by definition and $X \in \tau$ ...
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32 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
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0answers
35 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...
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2answers
31 views

Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...
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0answers
33 views

what is the probability that there is a string of k consecutive heads?

A coin is flipped n times. Assuming that the flips are independent, with each one coming up heads with probability p, what is the probability that there is a string of k consecutive heads? An answer ...
2
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4answers
92 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...