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

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2
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1answer
41 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (i) and (ii) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
1
vote
1answer
19 views

Dirac-Delta function representation - infinite sum involving trigonometric identities

Proof of the identity: $$\delta (x-x') = \sum_{n=0}^{\infty} \Big\{ \cos[n \pi(x-x')] - \cos[n \pi(x+x')] \Big\} $$ I can intuitively tell that this function is $\infty$ for $x=x'$, and zero ...
0
votes
0answers
40 views

Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix

Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix Here is my solution. Is it correct? So my thought was something along the lines of: $$\ker(A)\cap \mbox{Im}(A)\not ...
-10
votes
3answers
171 views

Please attempt to fault my proof of the continuum hypothesis [closed]

I have put this proof of the continuum hypotheses to both a Dr. of Maths and a Professor of Logic, and neither has demonstrated a flaw - although I doubt the professor (who shall remain nameless) ...
-2
votes
0answers
13 views

looking for polynomial with degenerate local (not gloabl) minimum [closed]

Can anyone give me a multivariate polynomial $f$ such that the origin is a degenerate critical point and also a local (not global) minimizer of $f$? Thanks in advance!
5
votes
1answer
44 views

For each pair of events A and B, find P(A|B) and P(B|A).

I've got a simple problem here, but I just want to ensure that I'm not losing a simple concept. Relevant equations: $$P(A|B) = \frac{P(A \cap B)}{P(B)},$$ $$P(A \cap B) = P(A) + P(B) - P(A \cup B)$$ ...
3
votes
2answers
36 views

Probability Independence - Determining if two sets are independent (drawing two cards)

I've got a few problems here that I feel pretty confident on. I am asking for confirmation on these answers. However, I am stuck on problem #3. Please let me know if you need more information. Two ...
2
votes
0answers
20 views

Lipschitz continuity of $f(x,y)=y^{1/2}$

Show that $f(x,y)=y^{1/2}$ (a) does not satisfy a Lipschitz condition on the rectangle $|x| \le 1$ and $0 \le y \le 1$. Answer: My book says that $\frac{f(x,y_1)-f(x,y_2)}{y_1-y_2}$ must be ...
0
votes
2answers
36 views

Is giving an explicit homeomorphism sufficient to prove that there exists a homeomorphism?

I am asked to prove that a square is homeomorphic to a circle. Now we can construct the homeomorphism explicitly by first having a bijection $\gamma$ that takes an arbitrary square in $\mathbb{R}^2$ ...
2
votes
0answers
17 views

Injective homomorphism from an ideal to a torsion free module over a Dedekind ring

Let $M$ be a nonzero torsion free module over a Dedekind ring $R$ and $I ⊂ R$ an ideal. Show that there is an injective $R$-module homomorphism $I → M$. My solution is to take a nonzero element ...
2
votes
1answer
17 views

Can you verify my solutions to these probabilities, given a coin is flipped 10 times?

I'm fresh into probability and I think it's important to ask a lot of questions since it seems probability really challenges your intuition. I'm working on the following problem, and have found a ...
1
vote
2answers
24 views

Show that $\bigcup_{n=1}^\infty A_n= B_1 \backslash \bigcap_{n=1}^\infty B_n$

Let $\{B_n\}$ be a decreasing set $B_1 \supseteq B_2 \supseteq B_3 \supseteq ....$ Define $A_n = B_1 \backslash B_n$ i.e. $A_1 = \varnothing, A_2 = B_1 \backslash B_2$ If we imagine $\{B_n\}$ as a ...
2
votes
2answers
25 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = ...
0
votes
0answers
19 views

Complex integral evaluation; I get the right answer, but one of my steps is a little fishy

The integral is $\int_{\gamma}\frac{1}{z^{2}-1}dz$ along the path $\gamma(t)=2e^{ti},\;t\in[0,2\pi]$ Which I attempt to do by parts: \begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz= ...
1
vote
1answer
24 views

Is the following language context-free?

I need to show whether the language $L_2$ is context-free or not where $L_2$= $\overline{L}$ such that L= { $a^nb^m$ : 0 ≤ n ≤ m ≤ 2n }. I am able to show that L is context-free , S­> aSb | aSbb | ε, ...
1
vote
2answers
52 views

Hint for prove $\forall n \in \Bbb N \sum _{i=1}^{n} \frac {1}{i!} \le 2 - \frac{1}{2^{n-1}}$?

I'm trying to prove the following inequality $\forall n \in \Bbb N$: $$\sum _{i=1}^{n} \frac {1}{i!} \le 2 - \frac{1}{2^{n-1}}$$ I'm doing it by induction. It's true for $P(1)$. So now I want to ...
0
votes
0answers
17 views

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k. I don't know how to start this one
0
votes
0answers
18 views

In a binary code, all coordinates partake in at least one non-information set

It is true that all non-MDS $(n,k)$ codes contains at least one $k$-sized coordinate subset that does not correspond to an information set (because all such subsets are information sets iff the code ...
-1
votes
1answer
34 views

Under what conditions this equation $F(x)=-c x^{n+p} - b x^n + (a-c)x^p -b =0$ has solutions?

Let $$ F(x)=-c x^{n+p} - b x^n + (a-c)x^p -b $$ where $$ (\mathscr{H})\left\{ \begin{array}{@{}ll@{}} x > 0, & \\ a > c > 0, & \\ b > 0, & \\ 0<p ...
2
votes
0answers
30 views

Suppose that $a, b ∈ N$ are relatively prime. Prove that, for any $k ∈ N$, $a^k$ and $b$ are relatively prime.

Note: I've asked this question before, but this one offers a proposed solution and I'm checking for verification. $a$ and $b$ are relatively prime if the greatest common divisor of them is $1$. I am ...
1
vote
1answer
26 views

Need help understanding algebra steps taken in proof of why an even minus an odd is odd

I don't understand the algebra used in the below example proof from my textbook. Where does the + 1 come from? Is it okay to just add 1 anywhere you want? Or is there some rule here or reason you ...
5
votes
1answer
88 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
1
vote
1answer
26 views

Probability proof and making sure I cannot make further simplifications to my answer

The question asks me to compute the probability (sums are ok) of the probability of having at least one of r cells empty with n>r balls thrown at the cells with equal likelihood of landing in any of ...
1
vote
2answers
33 views

Intersection of two column spaces

Let the matrices A and B be: $$A = \begin{bmatrix} 2 & 3 & 0 \\ 2 & 3 & 1 \\ 2 & 3 & 1 \end{bmatrix} and \space B = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 ...
1
vote
2answers
53 views

Nullspace and column space of invertible matrix

I want to show that the matrix $A$ $n\times n$ is invertible if and only if $N(A) = {0}$ and $C(A) = R^n$. So far, this is what I've got: Theorem: A is invertible $\implies N(A) = 0$ and $C(A) = 0$. ...
1
vote
1answer
27 views

Let $A$ a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$.

Let $A$ be a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$, where $q^{(n)}= \lbrace a \in A \mid \exists d \in A \setminus q\text{ such that }da \in q^n \rbrace$ and ...
0
votes
1answer
23 views

proof verification for natural deduction

Could someone please let me know if I got the following natural deduction correct for the following formula ...
1
vote
1answer
22 views

proof verification for truth table

Hi I wanted to know if I got this question correct. Below is the question and the truth table. I said that they were not equivalent as columns 4 and 5 are different. ...
0
votes
0answers
20 views

On a closed-form from particular values of the Riemann zeta function and divisor functions

I am looking if I can get a closed-form for an infinite series, but I don't know for what it is possible, without finish my computations (see my Question, below). From Applications (8.1 Infinite ...
1
vote
0answers
41 views

Function in $L^2(\mathbb R^n)$

I want to show that if we consider a polynomial $V$ in $R[X_1,...,X_n]$ such that $V$ is negative near infinity then $e^{-V(X)}\notin L^2(\mathbb R^n)$. This is what I tried to do: $V$ is ...
2
votes
3answers
66 views

Prove that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is constant

Suppose that the function $f:\mathbb{R}^2\to\mathbb{R}$ has first order partial derivatives and that $$\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0\qquad\text{for all ...
2
votes
0answers
54 views

Carother's “certainly” proof about measurable sets

Carother's Real Analysis text has the following Theorem. Can someone check if my proof is correct? $(i \Rightarrow ii)$ Let $E$ be a measurable. Let $I_k$ be open intervals, such that $$m^*(E) ...
2
votes
1answer
64 views

Errata for Mathematics: Its Content, Methods and Meaning?

I'm new here, so I hope this is the right place to post this! I am currently reading through the Dover edition of the textbook Mathematics: Its Content, Methods and Meaning, by Aleksandrov, ...
2
votes
0answers
35 views

Simple Vacuous Proof, Correct Approach?

I am doing some practice exercises as I am starting out on proofs but I noticed that though I am getting the correct approach between vacuous and trivial proofs, I am not doing it in the same format ...
0
votes
0answers
24 views

$a^n$ is decrescent for $a<1$, crescent for $a>1$ and constant for $a=1$

Well, for $a>0$, the sequence: $$a^n$$ is: decrescent if $a<1$ crescent if $a>1$ constant if $a=0$ In order to prove this, I did: $0<a<1 \implies 0<a^2<a$ Now, suppose it ...
1
vote
5answers
50 views

$x<y \iff x^{-1}>y^{-1}$

In order to prove the following: $$x<y \iff x^{-1}>y^{-1}$$ *for $x>0$ and $y>0$ I tried this: $$x<y\implies y-x>0$$ I have to prove that this, implies that ...
0
votes
1answer
24 views

Show that $E$ and its closure have the same limit points

My attempt goes like this: If $x$ is a limit point of $E$ then every neighborhood of $x$ contains at least one other point of $E$, call it $y$. Since both $x$ and $y$ belong to $E$, they also belong ...
0
votes
1answer
19 views

Proving that $\{[-n, n] \mid n \in \mathbb N\} \cup \{\varnothing, \mathbb R\}$ is a topology over $\mathbb R$

Prove that $$\tau = \{[-n, n] \mid n \in \mathbb N\} \cup \{\varnothing, \mathbb R\}$$ is a topology over $\mathbb R$ and compare it with the euclidean topology $\mathcal E_1$. My attempt ...
1
vote
1answer
33 views

A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
0
votes
2answers
36 views

Proving that $\overline{\overline{E_1 \setminus E_2} \setminus E_2} = \overline{E_1 \setminus E_2}$ in a topological space

Let $(X, \tau)$ be a topological space and $E_1,E_2 \subseteq X$. Prove that $$\overline{\overline{E_1 \setminus E_2} \setminus E_2} = \overline{E_1 \setminus E_2}.$$ Note: $\overline E$ ...
1
vote
1answer
32 views

Showing that complex functions with the same derivative on the unit disc differ by a constant

I have from class: If $U\subset \mathbb{C}$, convex and $f:U\rightarrow \mathbb{C}$ is holomorphic, then f has a primitive. My proof is: The fact that $D(0,1)$ is clearly convex and ...
1
vote
1answer
29 views

How to verify this relationship between area under the graph and the preimage?

Define $h : \mathbb{R} \to [0, \infty)$, Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary) I wish to show the following is true: $$H = ...
1
vote
2answers
56 views

Proving that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is.

This is what I'm proving: Let $F$ be a field. Let $\phi : F[x]\to F[x]$ be an isomorphism such that $\phi(a)=a$ for every $a\in F$. Prove that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is. ...
0
votes
0answers
34 views

Strictly increasing function $f$ such that $f(S)$ is connected. [duplicate]

Let $S=[0,1)\cup[2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected.Can i say that $f$ must be continuous ? According to me $f$ being monotone so only jump ...
1
vote
1answer
21 views

Compact Operator with Infinite rank Doesn't have a Closed Image

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. Claim: A compact operator $T$ which has infinite-rank has an image that isn't closed. I'm trying to prove this claim but I'm ...
1
vote
1answer
14 views

Inequality about absolute value of difference of supremums

Let $A$ and $B$ be nonempty bounded subsets of $\mathbb{R}$. Define $|A-B|=\{|x-y|: x\in A, y\in B\}$. I wish to prove the following claim. Claim: $$|\sup A-\sup B|\leq \sup |A-B|$$ Is the following ...
1
vote
4answers
94 views

Is this a proof for the Collatz conjecture

For this problem, which I believe is still unsolved, I was wondering what is wrong with this proof I thought of (probably is wrong somehow) https://en.wikipedia.org/wiki/Collatz_conjecture So my ...
1
vote
2answers
37 views

Difference between proving $\forall n(Q(n) \implies P(n))$ and $\forall nQ(n) \implies \forall nP(n)$

From what I can understand an almost identical proof structure is often used in a direct proof. To prove $\forall n(Q(n) \implies P(n))$ the common would be: (1.) Let $n$ be arbitrary. (2.) Assume ...
0
votes
0answers
17 views

Trivial solution of a matrix equation

Let $$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in M_2(\mathbb{Z}_N)$$ Suppose $\gcd{(\det{A},N)}=1$. Show that if $x,y$ are both not zero in $\mathbb{Z}_N$ then ...
1
vote
2answers
50 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...