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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1answer
38 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
2
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1answer
67 views

Help needed - Approximating functions with geometric integration and derivation

I've somehow managed to approximate some functions using cheap tricks as geometrically derivating the function and then geometrically integrating an easier equivalent of the derivative (see here for ...
3
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5answers
110 views

Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
0
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1answer
31 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
3
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2answers
121 views

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
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2answers
302 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
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1answer
132 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
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2answers
25 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
2
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1answer
58 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
2
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2answers
46 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
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2answers
41 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
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1answer
40 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
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0answers
25 views

derivative of the composition of two smooth map is the composition of derivatives

I can't type the LATEX so I uploaded the image. By Googling, I found that is the result of the pushforward, but there is no proof so I can't understand. ...
0
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1answer
48 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
3
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3answers
81 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
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2answers
81 views

Can we prove the formula for surface of revolution?

This is math. We like to prove things. However, proofs are rigorous processes (for a good reason) and are more than just "that idea looks like it could make sense". I've seen proofs for many ...
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1answer
92 views

Cayley's theorem : precision

I have the form of Cayley's theorem that doesn't say any more than : for every finite group G there exists n such that Sn has a subgroup which is isomorphic to G. Now I'd be interested in knowing ...
1
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1answer
36 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...
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2answers
25 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
3
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0answers
54 views

Verifying the Cofinite topology

I'm trying to provee that the cofinite topology is a valid topological space. I've defined it as $$C=\{\emptyset\}\cup\{S\subseteq X: X-S \quad\text{is finite }\}.$$ Now, clearly $\emptyset$ and ...
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3answers
62 views

How to go about proving that $\cos^2 x$ is everywhere differentiable?

My first line of reasoning was to try directly evaluating $$\lim\limits_{h \to 0}\frac{\cos^2 (x+h) - \cos^2 (x)}{h}$$ and showing such a limit existed for any x, but when $\cos^2(x)$ evaluates to ...
2
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3answers
90 views

Proving analytical statement, Intermediate Value Theorem

Let's define $f$ as a continuous function with $f:[0;2] \to \mathbb{R}$ and $f(0) = f(2)$. Now, I want to show that: $$\exists x_0 \in [0;1]:f(x_0) = f(x_0 + 1)$$ I tried to plot a few functions in ...
2
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1answer
81 views

A proof in naive set theory.

I am trying to use naive set theory to figure out a proof of the following statement: $$(x = u \land y = v) \to 〈x, y〉 = 〈u, v〉$$. What propositions should i use to prove this?
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2answers
74 views

Prove choosing $\lceil\frac{V}{2}\rceil$ vertices accounts for at least $\frac{3}{4}$ of edges

Give a polynomial-time algorithm that finds $\lceil\frac{V}{2}\rceil$ vertices that collectively account for at least $\frac{3}{4}$ of the edges in an arbitrary undirected graph. The algorithm I have ...
0
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1answer
21 views

Prove that a language is not regular.

I want to ask how to prove the following language is not regular using closure properties. I tried to use pumping lemma but I find the proof itself shaky. I'd appreciate if you can help. The ...
0
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2answers
23 views

Help proving this Proposition

For every natural number $n$, the integer $6^{2n+1}+8^{3n}$ is divisible by 7. I handled the base case quite well, but got stuck on the induction step. Any help would be greatly appreciated.
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2answers
57 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
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2answers
34 views

Using Direct Proof Technique prove that x*y is odd

I need help proving this statement. I'm studying for my final and I can't figure out how to finish it. P(x,y): If x and y are odd integers, then the product of x*d must also be odd. I know that $ x ...
3
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2answers
134 views

Law of excluded middle. Do we need it in proofs?

Quite often when I am making a natural deduction proof, and I have no fixed idea on how to continue. I find myself thinking: "lets start with some form of the law of the excluded middle (LEM) and ...
4
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5answers
234 views

Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$

I've seen the identity $\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ used here recently. I checked for proofs here ...
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2answers
55 views

Using Euclid's Algorithm prove..

Using Euclid's Algorithm prove that the fraction $\frac{24n+5}{18n+4}$ is in lowest terms. Is this solution going to be correct as a proof? Thanks for help!
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1answer
44 views

Question about the Least squares method

We have $n$ dots: $(x_1,y_1)\cdots (x_n,y_n)$. We know that if we use the Least squares method we will get a line $y=mx+b$ that giving the minimal value for the function $w=\sum_{i=1}^n ...
0
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0answers
15 views

Proof of update rule for perceptrons

$$w = \sum_n \tau^n\phi^nt^n$$ $$t^n = -1\text{ or }+1.$$ $\phi^n$ is a vector. $\tau^n$ is the number of times each vector $\phi^n$ has been presented and misclassified. It can be shown that ...
0
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1answer
41 views

Euler's Formula v-e+f=2

Assuming G is a planar graph with k components, I need to determine an equation relating vertices, edges, faces and components. It is given that when k=1, v-e+f=2. So from this I have gotten: ...
6
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2answers
244 views

Proof of continuity - (ε-δ) definition - Can anyone check this?

I've been trying to get my head around this problem for quite some time by now. I want to prove that $$f(x) := \left|\frac{x-1}{x^2+1}\right|$$ is continuous for $$x_0 = -1$$ Now, in order to prove ...
2
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1answer
17 views

Need help proving that $ fRg \Leftrightarrow fg = f $ on $ B^{n} $ to $ B $ if and only if $ f(b_1,…,b_n) \leq g(b_1,…,b_n) $

I'm trying to gather my thoughts for proving the following claim: For $ fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $ fRg $ if and only if for any input values $ b_1,...,b_n $, we ...
1
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1answer
34 views

Proof: If $F^3 = F$ then F is diagonalisable

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$. Show: F is diagonalisable. $F^3 = F$ is equivalent to $F^3 - F = 0$. Now I know that ...
0
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0answers
62 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
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2answers
29 views

Heuristics for Lipschitz equivalence

I'm studying for my finals in general topology and when I look at the definition of the Lipschitz equivalence of two metrics as: Let $d(x,y)$ and $d'(x,y)$ be metrics on a non-empty set $X$. We ...
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1answer
37 views

Inductive step assumption for all numbers up to $n$

I know that the inductive step should be "for all $n$ (if $P(n)$ then $P(n+1)$)" and NOT "if (for all $n$ $(P(n)$)) then (for all $n$ ($P(n+1)$))" - see this answer. But can it be like "if (for all ...
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1answer
40 views

Are these proofs correct?

I haven't formally learn how to do proofs, but I attempted some of these. It'd be great if you guys can check them and give me some pointers. Thanks!
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2answers
65 views

Ground Plan - Prove Fermat-Euclid's Totient Theorem with Lagrange's Theorem

If $\gcd(a,n) = 1$, then $a^{\phi(n)}\equiv 1\pmod n$. Here's a three-step proof. An integer a is invertible means there's some $a^{-1}$ such that $aa^{-1}\equiv 1 \pmod n$. By cause of Jones p84 ...
4
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2answers
188 views

Fermat's Little Theorem fails for composite instead of prime numbers.

I know Fermat's Little Theorem = Fermat-Euler's Totient Theorem when $n$ is prime. Elementary Number Theory, Jones, p83 writes if we simply replace p with a composite integer n, then the ...
2
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1answer
74 views

Backward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

(1) How can you preconceive to prove by contradiction? Prove by contradiction. Suppose $n$ is composite. This means there exists a divisor $d|n$ such that $1<d<n$. We are given that ...
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3answers
36 views

Question about the non-hausdorffness of the cofinite topology

I'm reading a proof about that disproves that the cofinite set $$O_c= \{\emptyset,X\}\cup\{S\subset X|\quad X-S\quad \text{is finite }\}$$ is hausdorff. It goes as follows : Let $U,V \subset X$ ...
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2answers
55 views

Ground Plan – Forward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

Lemma 5.3 - I omit proof here - Let p be prime. Then $x^2 \equiv 1 \, (mod p) \iff x \equiv \pm 1 \; (mod p)$ First we establish the result for the first two primes 2, 3. Then prove the result for ...
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1answer
92 views

Ground plan of Forward direction - Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Prove by contradiction. Thence suppose NOT $p\equiv 1 \; (mod 4)$. Thence 3 possibilities remain: $4|p, 4|(p - 2), 4|(p - 3)$. But $p > 2$ is prime, thence $4 \not | p$. (1) How can you ...
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1answer
20 views

Help proving a theorem in my textbook

If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational. For reference, an integer $n$ is a perfect square if $n=m^2$ for some $m \in \mathbb{Z}$. Any help proving this ...
2
votes
2answers
46 views

Help with a proof my professor gave my class

Let $x,y \in \mathbb{R}$ with $x<y$. There exists an irrational number $z$ such that $x<z<y$. My proof so far: Let $x,y \in \mathbb{R}$ and assume $x<y$. Then, by Theorem 11.8 (in ...
3
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1answer
174 views

Ground plan of Backward direction (<=) - Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Apply the identity $p-i \equiv -i \mod p$ for $i=1, \ldots$ to the pink factors $ \begin{align} \color{seagreen}{ (p-1)! } = 1\times 2\times\cdots\times \dfrac{p-1}{2} & \times \quad ...