Tagged Questions

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

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4
votes
4answers
92 views

Prove that $\sqrt{2} + \sqrt{3}$ is irrational. [duplicate]

Assume that $\sqrt{2}, \sqrt{3}, \sqrt{6}$ are all irrational. Prove that $\sqrt{2} + \sqrt{3}$ is irrational. So I know how to prove this using contradiction, and assuming that it is rational. But, I ...
3
votes
3answers
84 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. ...
1
vote
1answer
24 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 ...
0
votes
2answers
40 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 ...
0
votes
2answers
28 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 | ...
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 ...
0
votes
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 ...
1
vote
0answers
77 views
+50

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 ...
4
votes
3answers
42 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 ...
0
votes
1answer
31 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 ...
5
votes
0answers
61 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 ...
6
votes
0answers
120 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 $m(X)=1$. ...
5
votes
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 $X \in \tau$ ...
1
vote
0answers
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 ...
0
votes
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 $(a)$ ...
0
votes
2answers
26 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 ...
0
votes
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
votes
4answers
85 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 ...
1
vote
0answers
36 views

Exponential Function Limit Question

When I was first introduced to a derivation of the Taylor series representation of the exponential function here (pg 25): I noted the author, Dunham mentioning that the argument was non-rigorous. I ...
0
votes
1answer
25 views

Residue Calculus (Computing an Improper Integral)

Use residue calculus to compute the integral $\int_{-\infty}^{\infty}\frac{1}{(z^{2}+25)(z^{2}+16)}dz$ My solution If we add to the interval $I_{R}=[-R,R]$ add the semicircle $\gamma_{R}$ in the ...
2
votes
3answers
45 views

Applying Rouché's Theorem

Determine how many zeros of the following polynomial lie inside the circle $|z|=2$ \begin{equation} z^{5}+2z^{4}+z^{3}+20z^{2}+3z-1=0\end{equation} My Reasoning If we put $f(z)=z^{5}+2z^{4}$ and ...
0
votes
0answers
37 views

Proving that the Poisson random variable with mean $x$ is the Poisson random variable $X$ with maximum value of $\text{Pr}(X=x)$.

For any pair of positive numbers $\mu_1$ and $\mu_2$, let $X_1$ be the Poisson random variable with mean $\mu_1$, and $X_2$ be the Poisson random variable with mean $\mu_2$. Proof that $\text{Pr}(X_1 ...
0
votes
1answer
35 views

How to solve this *without* handshake theorem?

Suppose $45$ handshakes occurred in a room, how many people were in the room? Someone asked me this question and I was going to answer him using graph theory and my knowledge of the number of ...
-1
votes
2answers
24 views

Prove radius chord theorem without using congruent traingles

Suppose that $P(a,b)$ and $Q(c,d)$ are two points on the unit circle $x^2 + y^2 = 1$, and let $M$ be the midpoint of chord $PQ$. (Without using congruent triangles), prove that $OM$ is perpendicular ...
2
votes
4answers
101 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
1
vote
1answer
47 views

alternating sum of zeta functions minus one is one half

During my work on a different infinite series I happened to prove that $\displaystyle\sum_{k=2}^{\infty}(-1)^k (\zeta(k)-1)=\frac{1}{2}$ where $ ...
0
votes
1answer
48 views

Show that | and $\downarrow$ are the only binary connectives \$ such that {$} is functionally complete.

I've been reading and coping with van Dalen's Logic and Structure for a a few days. However, I've getting problems to solve his Exercise 6 from Ch 1 Sec 1.3 (p.28). In this exercise, van Daken asks ...
1
vote
1answer
33 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
2
votes
2answers
70 views

Alternate proof for $a^2+b^2+c^2\le 9R^2$

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful ...
1
vote
0answers
20 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
-1
votes
0answers
18 views

Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
1
vote
0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
0
votes
1answer
60 views

Two reasons why $\int^{1}_{0}f(x) \,dx$ exists?

Consider $f$ on $[0,1]$ defined as $f(0)=0$ $$f(x)=2^{-n}\quad \text{if}\quad 2^{-n-1}<x\le2^{-n},$$ for $n=0,1,2,3,...$ I'm looking for two reasons why $\int^{1}_{0}f(x) \,dx$ exists? One ...
0
votes
1answer
33 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
9
votes
4answers
163 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
1
vote
2answers
45 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
4
votes
0answers
113 views

Will this algorithm stop before time?

For every $n \in \mathbb N$, let's define $a_0 = 0$, $$\begin{cases} a_{i+1} = 2a_i + 1 \pmod {2^n}, &\text{if it never appeared before} \\ a_{i+1} = 2a_i \pmod {2^n},& ...
0
votes
1answer
76 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
0
votes
1answer
68 views

Prove that the graph $H = H_1\cup H_2 = (V_1\cup V_2,E_1\cup E_2)$ is connected.

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...
20
votes
1answer
445 views

Verify matrix identity $A^tD-C^tB=I$ on certain hypotheses

Given $n\times n$ real matrices $A,B,C,D$ such that: $AB^T$ and $CD^T$ are symmetric $AD^T-BC^T=I$ Prove that $A^TD-C^TB=I$ The solution I have come up with after a very long time is to consider: ...
3
votes
0answers
50 views

Is this irrationality proof correct?

Consider a non-square integer $n$. If its square root was rational, then we would have $$\sqrt n=\frac{a}{b}$$ for some $a,b\in\mathbb{Z}$ and so $a^2=nb^2$. But this is impossible, because $n$ is ...
3
votes
2answers
84 views

square root of 2 irrational - alternative proof

I have found the following alternative proof online. It looks amazingly elegant but I wonder if it is correct. I mean: should it not state that $(\sqrt{2}-1)\cdot k \in \mathbb{N}$ to be able to ...
1
vote
0answers
65 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
2
votes
4answers
141 views

Prove that the symmetric group $S_n$, $n \geq 3$, has trivial center.

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let ...
0
votes
1answer
28 views

Simplified Galois proof?

I have learned about Galois epoch-making proof that any polynomial of the fifth degree has no solution representable in terms of its coefficients. Can his proof be simplified and clarified in modern ...
6
votes
2answers
144 views

Alternative proof of Girard's theorem

I am looking for an alternative proof of Girard's theorem. The standard proof, which is almost trivial, relies too much on visualizing spherical triangles on the sphere. Is there a more algebraic ...
2
votes
2answers
79 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
3
votes
2answers
83 views

Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat: Characterize the natural numbers $n$ such that ...
0
votes
0answers
33 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
0
votes
1answer
74 views

Proving isomorphism between between a subspace and a quotient space

I've been thinking about this for a day or two now, and I think I've found a way to prove this, but am very unsure about how watertight this is: To be proven: Let $V$ be a vector space. If $V = U ...