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

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20 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
3
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0answers
15 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
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1answer
26 views

Find all primes $p$ for which $x^2+2x+4\equiv 0 \pmod p$ is solvable. Am I correct?

Getting ready for an exam, I would like to focus on the correctness of my solution, final results and assumptions, and would appreciate any comment regarding it or even additional ...
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2answers
28 views

Proving that $a^{b}$ is rational (Elementary number theorey) [duplicate]

Prove that there exist irrational numbers $a$ and $b$ such that $a^{b}$ is rational. What i tried Prove by contradiction I assume the statement For all rational numbers $a$ and $b$ such that ...
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0answers
38 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
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1answer
31 views

$\gamma=(\psi \implies \phi)$ is a tautology $\equiv \psi$ is a contradiction or $\phi$ is a tautology.

Prove that: If $\psi,\phi$ are formulas such that $\text{VAR$(\psi)$} \cap\text{VAR$(\phi)$}=\emptyset$. Then $\gamma=(\psi \implies \phi)$ is a tautology $\equiv \psi$ is a ...
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1answer
12 views

Show that a line is tangent to a circle in the extended complex plane.

The straight line $l$ in the extended-complex plane pasess through $2+i,2+2i$.The circle $C$ centered at $-1-2i$ with radius $3$. First, I find the parametrization of the straight line which is $$z = ...
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4answers
70 views

Proof of $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$

I was trying to prove $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$ and came across issues in translating (pertaining to what I did with $\emptyset$) and got through the proof but was doubting its accuracy so ...
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3answers
134 views

Prove that $\det(I-CD)=\det(I-DC) $

Let $C$ and $D$ be matrices such that $DC$ and $CD$ are square matrices of the same dimension. How can one prove that $\det(I-CD)=\det(I-DC)$? This is my approach to the question. I am not sure ...
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0answers
17 views

Generalized Associative Property (Proof Verification)

I am really confused about Associative property and Generalized associative property. I am not sure of my proof, and I have a feeling that it is not correct. Would be happy if someone can tell me what ...
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2answers
30 views

Proving a function $F$ is surjective if and only if $f$ is injective

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. Then we can define $F: P(Y) \rightarrow P(X)$ by \begin{align*} F(B) = f^{-1}(B) \qquad \text{for all} \ B \in ...
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0answers
36 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
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1answer
36 views

Check whether this is indeed a counterexample

Let $A,B \subset \mathbb{R}$; let $Q := A \times B$; and let $f: Q \to \mathbb{R}$ be bounded. The problem is to give a counterexample to the proposition that if the Riemann integral $\int_{Q}f$ ...
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2answers
35 views

If $\gcd(ab,c)=d$ and $c|ab$ then $c=d$

For all positive integers $a$, $b$, $c$ and $d$, if $\gcd(ab, c) = d$ and $c | ab$, then $c = d$. Need help proving this question, I know that $abx + cy = d$ for integers $x,y$ and that $c|ab$ can be ...
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0answers
24 views

Characterization of graphs of maps between smooth manifolds

Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ ...
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1answer
35 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
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1answer
16 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
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0answers
34 views

Least squares solutions of the linear system

I'm doing problems from old exams, and my solutions don't add up with the professor's solution. The problem is as followed: Find all least squares solutions of the linear system. I checked my ...
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2answers
40 views

Prove that no set can contain everything (or every other set)

Prove that there cannot exist a set that contains everything. Ill put my proof in the answer so please check it there. Also if there is a more creative way to do this(using the basic axioms) if it's ...
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0answers
45 views

Binomial theorem proof

I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own. I'm struggling with his proof of the binomial theorem, as summarized below. ...
2
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1answer
22 views

A surjective endomorphism (of a Noetherian ring) is injective.

The problem is stated as follows: "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective." Regardless of the ...
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3answers
61 views

Combinatorial Proof of a Simple Identity

Consider the following identity: $\binom n r = \frac n r \binom {n-1} {r-1}$ where $n \ge r \ge 1$. It's easy to supply an algebraic proof, but I'm looking for a combinatorial proof. I tried the ...
3
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1answer
28 views

If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$

The question is If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$. My attempt is that $b \equiv 0 \pmod a$ can be written $a\mid b-0 = a\mid b$ and the same with $c \equiv 0 ...
2
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1answer
29 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
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2answers
49 views

$\mathfrak{sl}(2)$ is a simple Lie algebra.

I am trying to prove that $\mathfrak{sl}(2,\Bbb C)$ is simple. Since this takes the $[x,y]=xy-yx$ matrix commutator bracket, this is clearly non-abelian. So to prove it is simple, we need only show ...
3
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2answers
70 views

Question about irrationality proof of $\sqrt{n}$

I'm talking specifically about a proof that I've found. I don't seem to get some parts of it. It states that if you take: $$\sqrt{n}=\frac{p}{q} \:\: \;\;p,q \in \mathbb{Z} $$ where $p$ and $q$ share ...
3
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2answers
35 views

Unique Linear Map- Linear Algebra

Let $E = {e_1, . . . , e_n}$ be a basis for $\mathbb{R}^n$ , and let $v_1, . . . , v_n$ be arbitrary vectors in $\mathbb{R}^m$. Prove that there is a unique linear map $T : \mathbb{R}^n \rightarrow ...
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1answer
40 views

Graphic proof of an inequality between sequence ratios

I would like to verify my proof for the following claim. Let $b_i$ be a positive decreasing sequence, $j<k$ two integers and $d$ a positive number. Prove that: $$ ...
0
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1answer
19 views

What am I doing wrong here? Showing $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$.

Show $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$ where $a$ is invertible. What I did is: If $\text{Ord}_{N}(a)|k$ it is obvious. Suppose $a^k\equiv 1 \pmod N$. Not let us assume by contradiction ...
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1answer
52 views

Limit of arithmetic means

If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}.$$ If $\lim s_n=s$, prove that $\lim \sigma_n=s.$ My proof: Let $t_n:=s_n-s$ ...
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1answer
32 views

If the Ideals Generated by the Coefficients of $f(X),g(X)$ are $R$, then so is the Ideal Generated by $f(X)g(X)$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the coefficients ...
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3answers
29 views

Power series with radius convergence $\leqslant 1$

Suppose that the coefficients of the power series $\sum a_nz^n$ are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1. Proof: Let radius of ...
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1answer
42 views

Proof that $A\cap\emptyset=\emptyset$

I'm trying to prove $A\cap\emptyset=\emptyset$. I've seen several proofs for this which all seemed to essentially go about proving it by noticing that $\emptyset\subset A\cap\emptyset$ by definition ...
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1answer
12 views

Proving with divisibility

I have never written any proofs (except high school geometry) in my life, so I'm not sure what exactly the proper formatting should be. Involving divisibility, the proposition states: Let $a, b,$ ...
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2answers
44 views

Proof about sum of convex polygon interior angles

I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own. I'm struggling with this proof (clipped from Google books): I follow him ...
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0answers
22 views

Let $R$ be a relation on $A$ and let $S$ be the transitive closure of $R$. Prove that $\text{Dom}(S) = \text{Dom}(R)$.

This is from "How To Prove It". The full exercise also asks to prove that $\text{Ran}(S) = \text{Ran}(R)$ but I was set from the outset on proving that $\text{Dom}(S) = \text{Dom}(R)$ first. Since the ...
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1answer
47 views

Proving well-ordering property of natural numbers without induction principle?

In Munkres, Topology, he has this way of proving the well ordering property for the natural numbers: He assumes he can work with the real numbers from the for the real numbers Then he defines an ...
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0answers
24 views

On properties of linear orders

I have a simple question. Let A={a,b,c,...} be a set and > a total strict order on $2^A$. Total strict order means that for any two subsets of A, say S and S', either S>S' or S'>S but not both. The ...
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0answers
31 views

Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
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1answer
43 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
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2answers
27 views

Rigorous Linear Transformation Proof

$T:V \rightarrow V$ We could also write: $T:V \rightarrow Im(T)$ The question tells us that $Im(T)=Im(T^2)$ It's intuitively obvious that this means that T then maps $Im(T)$ to itself so if you ...
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4answers
67 views

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...
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3answers
47 views

Why is this implication true?

$\dfrac{4x^2}{(1+x^2)^4} < 1 \to \dfrac{2\mid x \mid}{(1+x^2)^2} < 1$ . This is from a textbook i'm using (Advanced Engineering Mathematics, 10th ed., Kreyszig). It looks like the left hand side ...
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1answer
39 views

A problem from Finite Dimensional vector spaces

Problem : If $ M $ and $N$ are two subspaces of the vector space $V$ such that $\forall v \in V $ , $ v \in M $ or $ v \in $ (or both) . Prove that at least one of the is equal to $ V $ My ...
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0answers
32 views

Basic Set-Theoretic Properties from Halmos

I've been backtracking lately to make sure that I have a solid set-theoretic background before taking measure theory this fall. Here's a few facts I've come across today, and my attempted proofs. Let ...
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3answers
111 views

Proof that there are infinitely many prime numbers

I answered a question to prove that there are infinitely many prime numbers, but I'm not sure if my attempt is right. Can somebody help me to check if my attempt is right? I would like, if I am wrong, ...
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1answer
26 views

Verifying if these Cayley tables are from groups

For the first table I noticed that $ab = c \implies abb = cb \implies a = cb$ but in the table, $cb = d$, so this can't be a group For the second table, we have: $ab = c \implies (aa)b = ac ...
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1answer
14 views

$t > 0 $ is the least common multiple of $a, b$ (not both $0$) iff $a, b \mid t$ and $a, b \mid c \to t \mid c$

My attempt: Suppose $[a, b] = t =$ lcm of $a, b.$ By definition of lcm $a, b \mid t$. If $a, b \mid t$ and $a, b \mid c$, then $|t| \le |c|$ since $t$ is the smallest such integer. So, $t \mid c$. ...
1
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1answer
51 views

Easy proof for existence of Lebesgue-premeasure

In the lecture on measure theory I attended last semester, we had a sort of complicated technical proof for the existence of the Lebesgue-premeasure. However, I can't see why this easier argument does ...
2
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0answers
44 views

Artin's Algebra, Exercise 2.4.11. (1st edition)

I have been working through Artin's algebra book. Here's a simple exercise, but I want to make sure I am not missing anything important. Let $(G,\cdot)$ be a group and let $x,y \in G$ with orders ...