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

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0
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3answers
17 views

Proving if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$

Let $a,b,c\in \mathbb Z$. Prove that if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$. I get that sometimes this can acutally be false. Define ...
5
votes
3answers
51 views

Proving that if $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even.

Let $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even. My attempt: If one or two numbers of $a,b,c$ are even then we're done, so we'll have to show that at least one of them is even. ...
2
votes
3answers
49 views

Is this a valid way to prove that $\frac{d}{dx}e^x=e^x$?

$$e^x= 1+x/1!+x^2/2!+x^3/3!+x^4/4!\cdots$$ $$\frac{d}{dx}e^x= \frac{d}{dx}1+\frac{d}{dx}x+\frac{d}{dx}x^2/2!+\frac{d}{dx}x^3/3!+\frac{d}{dx}x^4/4!+\cdots$$ ...
0
votes
3answers
38 views

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) SO this is for abstract algebra and I am really struggling with this. Here are some of ...
0
votes
1answer
26 views

My proof that there are primitive roots modulo $p^2$

Let $p$ be a prime number. I'd like to prove that there are primitive roots modulo $p^2$. Could someone check this argument? Note that if $r\in\mathbb Z$ is a primitive root modulo $p^2$, it must ...
1
vote
1answer
12 views

Proof by induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$

Prove with induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$ Base case is trivial. Suppose that for a graph with $n-1$ vertices we have $|E|=\frac ...
3
votes
1answer
35 views

Axler LADR Exercise

The exercise is: Suppose $v_1, \ldots , v_m$ is linearly independent in $V$ and $w \in V$. Prove that if $v_1+w, \ldots, v_m+w$ is linearly dependent, then $w \in \operatorname{span}(v_1, \ldots, ...
2
votes
1answer
21 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
-1
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1answer
15 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
1
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1answer
32 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
1
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1answer
35 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
-3
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0answers
34 views

Prove the inequality between the arithmetic and geometric mean

Assume that for $x_1,...,x_n\geq0$ we let $G=(x_1x_2\dots x_n)^{1/n}$ and $A=(x_1+x_2+...+x_n)/n$. I would like to know if the following procedure leads to a proof of $$G\leq A$$ The equality is ...
0
votes
1answer
39 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
1
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0answers
35 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
0
votes
2answers
22 views

Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
1
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0answers
51 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
1
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1answer
26 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...
0
votes
0answers
21 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
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2answers
65 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
18 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
39 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
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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
30 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
29 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
29 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
61 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
65 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
39 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
28 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
34 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
9 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
33 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
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1answer
26 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
22 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
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1answer
28 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
1answer
18 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
221 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
16 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
1answer
19 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
14 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
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0answers
13 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
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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
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1answer
38 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
27 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
36 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
56 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
22 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
32 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 ...