For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

learn more… | top users | synonyms

0
votes
0answers
18 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
1
vote
2answers
50 views

$V$ is finite dimentional over field $K\iff$ field extension $L/K$ is finite

Let $L/K$ be a field extension and $V$ a non-zero vector space over $L$. Prove that: $V$ is finite dimensional over $K\iff V$ is finite dimensional over $L$ and $[L:K]<\infty$ for the first ...
0
votes
0answers
15 views

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
0
votes
3answers
36 views

Help me with proof concerning functions

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. We define $F: P(Y) \rightarrow P(X)$ by $F(B) = f^{-1}(B)$ for all $B \in P(Y)$. Proof that $F$ is injective if ...
1
vote
0answers
23 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...
0
votes
1answer
23 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
0
votes
1answer
26 views

Given the distribution of $X$ and $Y=-2\theta \ln X$. How is $Y$ distributed?

The pdf of $X$ is $f(x) = \theta x^{\theta-1},\enspace 0<x<1, \enspace 0<\theta<\infty.$ Let $Y=-2\theta \ln X.$ How is $Y$ distributed? My work: $$ \begin{align*} F(Y) = P(Y \leq y) ...
4
votes
0answers
28 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
2
votes
3answers
50 views

prove or disprove (discrete math)

This the question: Q: Prove or disprove the following statement. The difference of the square of any two consecutive integers is odd This is working step: let $m,m+1$ be 2 consective ...
1
vote
2answers
63 views

Is it true that $a$ can't be zero in the quadratic function $y=ax^2+bx+c$?

I read that for $y=ax^2+bx+c$ is a quadratic function where $a\neq0$, but is it true that $a$ really can't be zero? I think it is because if $a$ was zero, there wouldn't be a parabola. There would ...
2
votes
1answer
37 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
0
votes
1answer
26 views

Let $G$ be a connected graph, then $G$ is a tree iff $G$ has no cycles

Prove the following: Let $G$ be a connected graph, then $G$ is a tree $\iff$ $G$ has no cycles. $\Rightarrow$ If $G$ is connected and a tree then by the definition of tree it has no cycles. ...
2
votes
2answers
33 views

Integration by parts and $dx$ notation

Please overview this integral evaluation: $$ \int x^3 \arctan(x^2)dx = \frac{x^4}{4}\arctan(x^2) - \int \frac{1}{1+x^4}2x dx $$ Let's evaluate the right term: $$\int \frac{1}{1+x^4}\color{Blue}{2x ...
0
votes
1answer
8 views

Proof concerning indexed family of sets

Let $f: A \rightarrow B$ be a function. Let $I$ be a non-empty set, and let $\left\{U_i\right\}_{i \in I}$ be a family of sets indexed by $I$ such that $U_i \subset A$ for all $ i \in I$. Proof the ...
2
votes
1answer
30 views

Asymptotics of $\sum_{n\leq x}\tau_{k}\left(n\right)$

We define $\tau_{k}\left(n\right)$ to be the number of ordered $k$-tuples of positive integers with product equal to $n$. It is easily shown that this satisfies the recurrence relation ...
0
votes
1answer
25 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
2
votes
1answer
20 views

Given a graph on $n$ vertices find the maximum amount of edges so it can be colored with no monochromatic $K_m$

I invented a problem and I wanted to share :What is the maximum amount of edges a graph on $n$ vertices can have if it can be edge-colored with $k$ colors so that it does not have a monochromatic ...
1
vote
1answer
17 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
0
votes
0answers
6 views

Prove every integral ideal $J$ is identical with $\Bbb{J}_m$ for some $m$.

Prove every integral ideal $J$ is identical with $\Bbb{J}_m$ for some $m$. Suppose $J \neq \{0\} = \Bbb{J}_0$. By the least integer principle, there exists an $m \in J$ such that $rm \in J$ in $r ...
2
votes
2answers
220 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
0
votes
0answers
13 views

Convolution of negative binomial distribution w/ generalized binomial theorem

This is Exercise 3.1.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Show that $b^−_{r,p} \ast b^−_{s,p} = b^−_{r+s,p}$ for $r, s \in (0,\infty)$ and $p \in (0,1]$. ...
0
votes
0answers
13 views

Use the least integer principle to prove the following.

Least integer principle: Every non-empty set of positive integers has a least element. Using this fact, define $r$ to be the least integer for which $j - qk > 0$ where $j, k \in \Bbb{Z}$ ...
0
votes
1answer
13 views

Proving inverse implication by conversion

I have proven logically that the inverse of an implication is true if and only if the converse of said implication is true (as shown below). proposition 1: k has same parity as 2j proposition 2: k ...
0
votes
0answers
12 views

Prove for each pair of integers $j, k : k > 0$, there exists a $q : j - qk > 0$

Prove for each pair of integers $j, k : k > 0$, there exists a $q : j - qk > 0$. I began by writing out all three cases, i.e. $C_1 \to j > k$, $C_2 \to j = k$, and $C_3 \to j < k$. ...
1
vote
0answers
22 views

Does this reasoning about fourier analysis make sense?

I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t. $$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$ Here is my attempt: Assume there are ...
0
votes
1answer
37 views

prove $\lim_{x\to 0} f(x) = 0$ using epsilon delta

prove using Epsilon Delta that $\lim_{x\to 0} f(x) = 0$, where $f(x) = \left\{ \begin{array}{l l} \;\;\; \sqrt6 \;x & \quad \text{if } {x \in \mathbb{Q}}\\ -\sqrt6 \;x & \quad ...
0
votes
1answer
25 views

Integral of nonnegative function on plane domain gives a negative result, what is wrong?

Given an area $D: x \ge y, 0 \le x \le 1, y \ge 0$. $$ f(x,y)= \begin{cases} 2, & (x,y) \in D,\\ 0, & \text {others}\end{cases} $$ For this area $D_1: x+y \le 1, 0 \le y \le x$, I'm ...
1
vote
4answers
57 views

Prove that if $a<b$, then $-b<-a$

Prove that if $a<b$, then $-b<-a$ I'm a bit lost in this one, this is what I did: First case: $0<a<b$ $|a|<|b|$, so $|-a|<|-b|$ Since both are negative and $|-b|$ is greater ...
2
votes
2answers
35 views

Is $\sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous?

$f(x) = \sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous because: Proof: $f(x)$ is a continuous function on $[0, 1]$, which is a closed interval, so $f$ is uniformly continuous on ...
1
vote
1answer
35 views

Clarification on Cantor Diagonalization argument?

My book is Discrete Mathematics and its Applications. This is its section on Cantor's Diagonalization argument I understand the beginning of the method. The author is using a proof by ...
2
votes
3answers
55 views

How to approach this proof problem, what proof to use, what assumption to use?

This is a problem from Discrete Mathematics and its Applications Here is the definition of rational that my book uses Usually when I approach this type of a problem, I can find a type of proof to ...
0
votes
0answers
20 views

Would it be necessary to have another proof within the proof by cases in this problem?

This is a problem from Discrete Mathematics and its Applications I am using Proof by Cases. This is my book's definition on it. Here is my work so far I tried to leverage without of generality ...
1
vote
0answers
31 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
2
votes
2answers
42 views

How can we show that “almost surely” equal random variables have the same distribution?

How can we show that "almost surely" equal random variables have the same distribution? We know $X =\text{(a.s)} Y$. What I have so far: $$\begin{align*}\implies& P(X = Y) = 1 \\ ...
0
votes
1answer
44 views

Proof $\sum{ k{ x }^{ -k }=\frac { x }{ { (x-1) }^{ 2 } } }$

As the title says, I want to prove the following: $$\sum {k{x}^{-k}=\frac{x}{{(x-1)}^{2}}}$$ But I think I am doing something wrong. I start from the following: $$\sum{x^k} = \frac{x}{1-x} \implies ...
0
votes
0answers
12 views

Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
0
votes
0answers
31 views

$T(n) = T(n/3) + T(2n/3) + cn$ - recursion tree with constance $c$

I have a task: Explain that by using recursion tree that solution for: $T(n)=T(\frac n3)+T(\frac 2n3)+cn$ Where c is constance, is $\Omega(n\lg n)$ My solution: 1. Recursion tree for $T(n)=T(\frac ...
1
vote
0answers
31 views

Upper triangular matrices $UT_n(\mathbb K)$ is a matrix group: is my proof correct?

I am solving some exercises in Tapp's matrix groups for undergraduates and would be very grateful if someone could check my work (it's exercise 4.12): A matrix $A\in M_n(\mathbb K)$ is called ...
0
votes
4answers
55 views

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ So far I've got: $a<b$ $a^2<ba$ $a<\sqrt{ab}$ And: $a<b$ $a+b<2b$ $\frac{a+b}{2}<b$ So I need to prove ...
1
vote
2answers
30 views

The set of all exterior points is an open set

Let $S$ be a subset of $R$. Then the set of all exterior points of $S$ is an open set. My proof is as follows: For any element $x$ in $\operatorname{ext}(S)$ (the set of all exterior points ...
0
votes
2answers
29 views

Next step to take in this proof by contradiction?

This is a problem from Discrete Mathematics and its Applications Here is my work so far It's similar to this other question I had Next step to take to reach the contradiction?. I am assuming ...
2
votes
3answers
46 views

Can someone verify my direct proof that if A is a subset of B, AU B = B?

This is a problem from Discrete Mathematics and its Applications I am trying to use a direct proof to do this problem. Here is my book's explanation/section on direct proof Here is my work so ...
2
votes
2answers
66 views

Determining $\gcd(94, 27)$

I want to determine $\gcd(94, 27)$. Using the Euclidean algorithm, I got \begin{align} 94 &= 27 (3) + 13 \\ \implies 27 &= 13 (2) + 1 \\ \implies \;\;2 &= 2 (1) \end{align} Does this ...
0
votes
1answer
69 views

Big O Proof by Contradiction

Question: Use a proof by contradiction to show that $5^n$ is not $O(3^n)$ NOTE: This is homework, please don't provide an answer, just want to know if I am on the right track. My Attempt: ...
1
vote
1answer
39 views

Adjacency matrix and existence of triangle

Show that a graph $G$ contains a triangle (1) if and only if there exist indices $i$ and $j$ such that both the matrices $A_G$ and $A^{2}_{G}$ have the entry $(i, j)$ nonzero, where $A_{G}$ is the ...
0
votes
2answers
34 views

Trouble solving this induction problem

Show that, for every $n\ge2$, $3^n >n(n-1)$. Well, I started by showing the base case ($n = 2$): $3^2 > 2$ Now, for $n+1$: $P(n)\Rightarrow P(n+1)$ $$3^{n+1} > (n+1)n$$ My ...
0
votes
0answers
24 views

Verify my proof: if $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive.

Could someone verify my proof and my writing? Proposition: If $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive. Definition 1: A binary ...
3
votes
1answer
26 views

Is this a correct way to prove this?

I've just looked at this question and sketched a way to do it my head. When I looked at the answer it looked slightly more complicated than the way I did it so I just wanted to check whether this is a ...
3
votes
0answers
43 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
1
vote
1answer
39 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...