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

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0
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2answers
15 views

Mistake in proof of sum of divisors function $\sigma(n)$

The proof derives the correct result, but I cannot see how the first equality is correct. To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$ This is the first step in the proof: $$\sum_{1\leq ...
1
vote
0answers
20 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
0
votes
0answers
13 views

Solution verification dimension of a union of two sets

Let $V$ be a vector space. Let $W_1$ and $W_2$ be subspace of $V$. Then $W_1+W_2 = \{a+b~|~a\in W_1 \text{ and } b\in W_2\}$. (i)Prove that $W_1+W_2$ is a subspace of V. To show its a ...
1
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2answers
26 views

Prove that the sum of two positive real numbers is equal or greater than the square root of their product.

Trying to prove this: A and B are positive real numbers. A + B ≥ √ AB  This is what I wrote: Proof by Contradiction A + B < √ AB  (A + B)2 < AB A2 + AB ...
3
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0answers
25 views

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ...
1
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1answer
27 views

If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
6
votes
5answers
43 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
1
vote
1answer
24 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
1
vote
1answer
20 views

Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
1
vote
1answer
22 views

Galois group of $x^p-x-a$

$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$. First I showed that this is an irreducible polynomial and has no multiple roots. ...
3
votes
1answer
24 views

An identity for $J_n(x)$

It was much easier and faster to upload the image of my proof. I got it down to where the question marks are. But I can not seem to figure out how to get the anti-derivative to evaluate the integral. ...
5
votes
1answer
32 views

If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
0
votes
0answers
29 views

Solution Verification: Prove in detail that the open rectangles in the Euclidean plane form an open base

I want some verification and/or some polishing on my proof. However if it is good, please let me know (I think this is highly unlikely to happen). Problem. Prove in detail that the open rectangles ...
1
vote
1answer
30 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
3
votes
1answer
36 views

Deducing that the inverse of a permutation matrix is its transpose

I would like to verify that my proof below is sound. Let $A\in P$ where $P$ is the set of all permutation matrices (only one 1 in each row and column). Also, let $(A)_{ij}$ denote the entry of $A$ in ...
0
votes
0answers
18 views

Question about the deduction of the quotient ring $R/I$

Yesterday we deduced on class how quotient groups were deduced and well defined. Let $R$ be a ring and $I$ an ideal of $R$. My professor proved us that the multiplication operation $$R/I \times R/I ...
11
votes
1answer
165 views

A proof involving the Euler phi function

Problem: Let $\varphi$ be the Euler phi function, where for any $n \in \mathbb{Z^+}$, $\varphi(n)$ is the number of positive integers less than $n$ that are relatively prime with $n$. ...
1
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0answers
27 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
0
votes
0answers
14 views

Finding the range of a cannonball- proof verification.

I asked such a question before but I do learn best by mistakes and corrections.(I didn't fully understand it yet.) I could really use your verification: A cannonball is being fired with a velocity of ...
0
votes
0answers
37 views

proof that $\frac{e^{-t}}{2}(t^2+2t+2)\le1$ for $t\ge0$

Show that $\forall t\ge0,x\le1$ where $$x=\frac{e^{-t}}{2}(t^2+2t+2),t\in\mathbb{R}.$$ My proof we have $x=\frac{(t^2+2t+2)e^{-t}}{2}$ then ...
2
votes
1answer
34 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...
2
votes
1answer
17 views

Is this simple proof that the Frobenius endomorphism of an elliptic curve defined over $\mathbb F_q$ is surjective valid?

I am quite sure that the following "proof" is flawed, but I don't see why: Let $E$ be an elliptic curve defined over $\mathbb F_q$. Since $E$'s ideal is generated by a polynomial in $\mathbb ...
2
votes
0answers
35 views

Proving Finiteness

For any set $X$, if $\cup X$ is finite, then $P(X)$ (power set of X) is finite. For any Transitive set $X$, if $P(X)$ is finite, then $\cup X$ is finite. I'm a little confused with these because ...
3
votes
1answer
41 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
1
vote
1answer
35 views

Use Least Upper Bound to show that $\mathbb{R}$ is completee

Use Least Upper Bound to show that $\mathbb{R}$ is complete. The following is the proof I did, and it's slightly different from what I see from the book. Can someone check if I'm missing ...
2
votes
1answer
43 views

Prove that $\lim_{x \to a} \big[ f(x)+g(x)\big] = p+q$

Let $f: A \to \mathbb{R}^m$ and $g: B \to \mathbb{R}^m$ be functions such that $A,B \subseteq \mathbb{R}^p$, $a \in \mathbb{R}^p$ and let $\beta$ be a fixed number. Furthermore let $f(x) = ...
0
votes
1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
0
votes
1answer
13 views

Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
0
votes
1answer
25 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
6
votes
4answers
81 views

Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.

My problem in my solution is that I don't know if the operations I apply on congruence modulo n are admissible. I could really use some guiding: Attempt: Let there be $p\ne 3$ fulfilling the ...
0
votes
1answer
59 views

Verification of solution of a contest problem with a limit of nested radicals

They gave me 0 points for this problem. I think it's unfair. What do you think of this proof, is it correct? $\lim\limits_{n\to\infty} \underset{2n\text{ roots ...
2
votes
1answer
63 views

Using Fermats prime numbers to prove that there is infinitely many prime numbers

A Fermat number $F_n$ is of the form $F_n = 2^{2^n} + 1$ Furthermore, $F_n = 2 + F_0F_1F_2......F_{n-1}$ Now I already proved that if $n \neq m$ then $\gcd(F_n,F_m) = 1$ Here is the proof Without ...
2
votes
0answers
33 views

Argument verification fermat divisors.

any prime divisor of p is of the form then p = k $2^{n + 1}$ + 1 for n $\geq$ 2. We can use the result that Any divisor of $F_n$ is of the form q = k * $2^{n + 1}$ + 1 (*) Given that $F_n$ = ...
9
votes
3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
2
votes
2answers
30 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
2
votes
1answer
16 views

Proving $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$, where $A,B\subseteq S$

Prove that $\alpha\colon S\to T$ is one-to-one if $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$. Book solution: Assume that $\alpha(A\cap B)=\alpha(A)\cap\alpha(B)$ for every pair of subsets $A$ and ...
1
vote
2answers
64 views

Is my proof of a seemingly trivial question involving infinite sums correct?

Given that $\sum_{n=0}^{\infty}a_n^2<\infty$ and $\sum_{n=0}^{\infty}b_n^2<\infty$, where $a_n$ and $b_n$ are sequences of real numbers, I'm trying to show that ...
1
vote
1answer
49 views

Probability of same birthday

I think I solved this problem but I would like to know if I am right or wrong, I am not quite sure. We assume that the year has 365 days and the birthdays are uniformly distributed. We want to find ...
2
votes
1answer
31 views

Proving that if $S$ has an infinite subset then $S$ is infinite

Definition$\quad$ A set $S$ can be defined as infinite if there exists a mapping from $S$ to $S$ that is one-to-one but not onto. Otherwise, $S$ is finite. Problem: Using the definition of ...
2
votes
2answers
18 views

Proving that a positive derivative means the function is smaller “to the left” and larger “to the right” for certain values

I was trying to prove that if $g$ is differentiable on an open interval $I$ with $a\in I$ and $g'(a)>0$ then we can find $x<a$ for which $g(x)<g(a)$ and $y>a$ for which $g(y)>g(a)$, I ...
6
votes
1answer
36 views

Is $f(x)$ reducible if $f(a)=0$

I am confused about this seemingly trivial question: If $f(a) = 0$ for some $a\in D$, then when is $f(x)$ reducible in $D[x]$? ($D$ is an integral domain). My answer: Always. Let $f(a)=0$. ...
0
votes
1answer
29 views

$n$ divides $a_1 - a_2$ as well as $b_1 - b_2$. Show that $n$ divides $a_1b_1 - a_2b_2$.

I keep arriving at $a_1b_1$ and $a_2b_2$ having the same sign if I multiply the equations $a_1 - a_2 = nk$ and $b_1 - b_2= np$ times each other. They must be opposite signs so that I can say that $n$ ...
1
vote
2answers
29 views

Which of the properties, Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, Transitive, Linear, does F satisfy?

Let $S={(n,m) ∶n,m∈Z^+}$. Define the relation F on S by ${(n,m),(i,j)}∈F$ if and only if $nj=mi$. In other words, let $F = {((n, m), (i, j)) ∈ S × S: nj = mi}$. Proof F is reflexive: Show that for ...
1
vote
1answer
24 views

displacement of water

I just read the following puzzle on puzzling SE and it made me wonder, is there a proof which can be formulated to see if an object of any size and weight would cause more displacement inside the boat ...
1
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0answers
31 views

Two statements about partial limits and intervals

Let $(a_n)$ be a sequence and $I=(a,b)$ an open interval such that every $L\in(a,b)$ is a partial limit of $(a_n)$. Decide whether or not the following statements are true: $\{a_n | n \in ...
1
vote
0answers
14 views

Verify combination of disjoint subsets $C$ and $D$ is onto

Let $C$ and $D$ be disjoint subsets of set $A$ and $f:C→B$ and $g:D→B$. Define a function $h(x)$ as follows: $$ h(x)=\left\{ \begin{array}{c} f(x) \textrm{ if } x∈C \\ g(x) \textrm{ if } x∈D ...
2
votes
1answer
18 views

Finding upper bounds of a set

Suppose $A = \{ x \in \mathbb{R} : x + \frac{1}{x} < 5 \} $. I want to find 2 upper bounds of $A$. My intuition tells me that $5$ is indeed an upper bound. To see this, I want to show that for ...
3
votes
1answer
32 views

Are there general guidelines to make “assumptions” when proving limits?

I am studying the definition of the limit using Paul's Online Notes When proving the following limit (Example 3) $\lim\limits_{x \to 2} x^2+x-11 = 9$ At one point he assumes: $|x+5| < K$ He ...
1
vote
2answers
33 views

How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$.

How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$. What I did: I have $A:5$ $B:2$ $R:2$ $C:1$ $D:1$ If the words must end in a ...
3
votes
1answer
20 views

Proving something is not a Normal Extension

Let $M = \mathbb{Q}(\sqrt{3}, i\sqrt[4]{5})$ be an extension of $\mathbb{Q}$. Then work out the basis of $M$ over $\mathbb{Q}$ and show that the extension $M/\mathbb{Q}$ is not a normal extension. So ...