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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2
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2answers
85 views

Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
2
votes
1answer
23 views

Reduce all cases to $x \to 0^{+}$ and $f(x),g(x) \to 0$ before proving L'Hôpital's Rule

Hypotheses: I. $f$ and g differentiable on $(a,b)$ and continuous on $(a,b]$, II. $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 \quad $ or $\pm \infty$ III. $\lim_{x \to a} f'(x)/g'(x)$ exists IV. ...
2
votes
0answers
203 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
2
votes
3answers
242 views

Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$

I haven't been able to do this exercise: Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X ...
2
votes
1answer
79 views

Proofs involving sequential limit

Let $S$ be the domain of the function $f$. Suppose $S=\left\{\frac{1}{n}: n\in\Bbb N\right\}$. Show $\lim_{x\to0}f(x)=L$ iff $\lim_{n\to\infty}f\left(\frac{1}{n}\right)=L$. Idea: I want to say, let ...
2
votes
2answers
118 views

Proof that this is independent

Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set. I can think of the long way which is to differentiate this and put the differentiations into a matrix and row ...
2
votes
1answer
119 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
2
votes
3answers
248 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
2
votes
1answer
841 views

lim sup inequality proof - is this the right way to think?

I have tried to read many proofs of this but I'm not sure I get it, so please bare with me. Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
2
votes
2answers
95 views

L shaped region

Prove, using the well-ordering principle, that, for all $n\geq 1$, an $\mathsf{L}$-shaped space with two sides of length $2n$ and four sides of length $n$ can be tiled using some number of 3 square ...
2
votes
4answers
128 views

Is a contradiction enough to prove a set equality to $\varnothing$?

An exercise asks me to prove the following: $$A \cap (B-A) = \varnothing$$ This is what I did: I need to prove that $A \cap (B-A) \subseteq \varnothing$ and $\varnothing \subseteq A \cap (B-A)$ The ...
2
votes
2answers
141 views

Sum of Sines Interval [duplicate]

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? How is it possible to show for integer $m$: ...
1
vote
7answers
105 views

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$ I know this is true because any even number that is squared will be even, is it also true than any even number ...
1
vote
1answer
61 views

Application of Cauchy Integral

Let $f$ be holomorphic in $\{|z|<R\},$ where $R>1.$ Show: $\begin{align}f(z)= i\text{Im}f(0) +\dfrac{1}{2\pi} \int^{2\pi}_{0} \dfrac{e^{it}+z}{e^{it}-z} \text{Re}f(e^{it})dt, \ \forall |z| ...
1
vote
1answer
187 views

Number of roots of a polynomial over a finite field

For any $g$ in $\mathbb{Z}/p\mathbb{Z}[x]$ prove that the degree of $f = \gcd(x^p - x, g(x))$ is exactly the number of distinct roots of $g$ in $\mathbb{Z}/p\mathbb{Z}$. My main problem is that I ...
1
vote
1answer
178 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
1
vote
0answers
30 views

Prove the von Staudt-Clausen congruence of the Bernoulli numbers

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
1
vote
1answer
41 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 $T$ ...
1
vote
2answers
37 views

Not all ideals are finitely generated

Let $R=\{a_0+a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, a_o \in \mathbb{Z}, n\in \mathbb{Z}_{\geq 0} \}$ and $I=\{a_1X+...+a_nX^n \ | \ a_1,...,a_n \in \mathbb{Q}, n\in \mathbb{Z}^{+} \}.$ ...
1
vote
4answers
500 views

proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$

Question: Prove that the sum of the binomial coefficients for the nth power of $(x + y)$ is $2^n$. i.e. the sum of the numbers in the $(n + 1)^{st}$ row of Pascal’s Triangle is $2^n$ i.e. prove ...
1
vote
1answer
159 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://www.math.uga.edu/~pete/2400calc2.pdf. This theorem can be ...
1
vote
3answers
147 views

Arithmetic and geometric mean

I need to prove that for $a=\frac{x+y}{2}$ and $g=\sqrt{xy}$, following statments are true or false: For $x\not =y,a>g$ and $x=y, a=g$. I have no idea how to do this, so any help is welcomed. ...
1
vote
1answer
161 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
1
vote
1answer
360 views

Combinatorially prove something

So i'm not sure at all how to prove things using a combinatorial proof. Where to do i start? What do i need to think about etc. For example how would i prove $$\sum_{i=0}^n {n \choose i} 2^i = 3^n ...
1
vote
1answer
414 views

Find a bijection from $(A^B)^C$ into $A^{B \times C}$ [duplicate]

Possible Duplicate: How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers? Notation: Let A and B be sets. The set of all functions $f:A \rightarrow B$ is denoted by $B^A$. Problem: ...
0
votes
2answers
72 views

Proof of a point's existence in an open interval

Well let us begin consider a set $A$ $$A = \{a \le x \le b \space | \space f(x) > 0 \}$$ Lets take $\alpha = \sup A$ and $\beta = \inf A$ What we must do is prove that $\alpha = d$ and ...
0
votes
1answer
43 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 ⇒ ...
0
votes
1answer
39 views

Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function? [closed]

Let $A=B=\mathbb N$. Relation $R$ is: $(a,b)\in R$ iff for some $q \in \mathbb Z$ we have $a=5q+b$ Given a relation, show that it's a function. To Show: $\forall a \in A \ \exists b \in B$ such ...
0
votes
3answers
35 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
votes
5answers
82 views

Series and sequences convergence with a certain condition.

Let $\sum_{n=1}^\infty (a_{n})$ converge. Let $\{n_{k}\}$ be a subsequence of the sequence of positive integers. For each $k$ define $b_{k}=a_{n_{k-1}+1}+...+a_{n_{k}}$ where $n_{0}=0$. Prove that ...
0
votes
1answer
45 views

Prove that there is a 1-1 correspondence between the set of subgroups of $\mathbb{Z}/N \mathbb{Z}$ and the set of the positive divisors of $N$

Im interested in the above Proof, is because I have the intiuition that it is not true at all, because for example, all the primes have exactly 2 positive divisors 1 an themselves, How Can I prove or ...
0
votes
1answer
34 views

Show that are logically equivalent

Can you answer me , please. 1- Show that (p→r)∧(q→r) and (p∨q)→r are logically equivalent 2- show that (p → r) ∨ (q → r) and (p ∧ q) → r are logically equivalent without truth table .
0
votes
1answer
43 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
0
votes
1answer
70 views

Proofing a Reachable Node Algorithm for Graphs

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
0
votes
3answers
119 views

How to prove if $n$ is prime and $n | a^2$ then $n | a$?

My professor assigned this for homework but I don't understand how to connect the dots. He suggested using the fact that $\gcd (x,y) \cdot \operatorname{lcm} (x,y) = xy$ but I'm not sure how that's ...
0
votes
1answer
53 views

Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
0
votes
4answers
77 views

Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$

Determine if $g$ is injective; surjective; bijective. Question on a recent test regarding one-to-one and onto functions. Was very difficult for me, could not even begin to answer either. This is ...
0
votes
0answers
57 views

Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
0
votes
1answer
59 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
0
votes
4answers
70 views

Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$

Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$ Extra question in my proofs homework, similar to another question I posted, but not exactly the same I don't think. ...
0
votes
2answers
50 views

Need help with a math proof

Any help would be greatly appreciated. Let $m,n,p,q \in \mathbb{Z}$. If $0 < m < n$ and $0 < p < q$ then $mp < nq$.
0
votes
2answers
105 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
0
votes
1answer
118 views

Natural deduction proof of a simple formula.

I am trying to prove the following formula using only natural deduction system: $$\vdash (A \supset (A \supset B)) \supset (A \supset B),$$ and this, according to natural deduction rules leads to $$A ...
0
votes
3answers
102 views

Math real numbers analysis.

Given a real number $x$, and a natural number $N\gt 1$. Consider the numbers : $0, x−\lfloor x\rfloor,2x−\lfloor2x\rfloor,\ldots,Nx−\lfloor Nx\rfloor.$ Show that some pair of these numbers differs ...
0
votes
2answers
168 views

Proof for $\epsilon$ closeness to $\sqrt{2}$ for all values of $\epsilon$

I am trying to understand the proof above, Proposition 7, given by the author, but I got stuck at some points. My explanation: Now, suppose for contradiction there is an $\epsilon > ...
0
votes
1answer
158 views

how to prove this scheduling problem

I need some hints for proving the correctness/optimality of the below homework problem. It is a task-schedulding problem with deadlines and penalties. There are n tasks, each of which has a deadline ...
0
votes
6answers
1k views

Can I prove (if $n^2$ is even then $n$ is even) directly?

I want to prove that if $n^2$ is even then $n$ is even directly without using the contrapositive or the contradiction methods.
-1
votes
1answer
88 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
votes
3answers
660 views

How many ways can $5$ rings be placed on $4$ fingers?

I've been trying to solve this problem and I am kind of struggling with it and with other combinatorics problems. Could you check and see if i did it right? Given problem: How many ways can 5 ...
-2
votes
1answer
808 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...