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

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24 views

Prove $α · β ≤ α · γ$ and $α^ β ≤ α^ γ$ for any three cardinals, where $ β ≤ γ$.

This is what I did: a. let $|A|=α, |B|=β, |C|=γ. |B|≤|C|$ and therefore there is an injection $f: B \to C$, such that $f(b)=c$ for some $b \in B, c \in C.$ Upgrading this function to ...
1
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1answer
48 views

Prove that a function is continuous at $x = x_{0}$ using the $\delta - \epsilon$ definition

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$ $\forall \epsilon > 0$, $\exists \delta = ?$ ...
1
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0answers
38 views

Module is zero if localization at associated primes is zero

Let $A$ be a Noetherian ring and $M$ an $A$-module. I want to show that $M=0$ if $M_P = 0$ for each $P \in \text{Ass}(M)$. Here is my attempt at a solution: Assume for a contradiction that $M ...
3
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2answers
60 views

Series convergence question from baby Rudin.

Suppose, $a_n>0$, $s_n=a_1+\cdots +a_n$, and $\sum a_n$ diverges. Prove that $$\frac{a_{N+1}}{s_{N+1}}+\cdots\frac{a_{N+k}}{s_{N+k}}\ge 1-\frac{s_N}{s_{N+k}}$$ and deduce that ...
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1answer
51 views

Prove that function $f$ is continuous at $x = x_{0}$

In class we're given the following definition about continuity, and I want to apply this definition to the problems that follow: $f$ is continuous at $x_{0} \in \mathrm{dom}(f)$ if $\forall x_{n} \in ...
2
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1answer
65 views

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$

Prove that $$f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$$ is continuous at $0$ and discontinuous everywhere else Proof: ...
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1answer
33 views

Proof of theorem in infimum and supremum

The first line is the statement that I want to prove. Let A and B be bounded non-empty subsets of R. Can someone please tell me does my proof (especially second last line) of this question valid or ...
1
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1answer
29 views

Spectral norm upper bound for covariance matrix

Let $\|\cdot\|_2$ be the spectral norm. Let $x_1,\dots,x_n$ be i.i.d. draws from $N(0,S)$. Let $\lambda_1,\dots,\lambda_n$ be some real numbers. Is it true that $$\|\sum_{i=1}^n \lambda_i x_i ...
2
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2answers
54 views

A rank-nullity theorem between $\mathbb Z^n$ and $\mathbb Z^k$ [duplicate]

I think this is correct: If $\phi:\mathbb Z^{n}\to\mathbb Z^{k}$ is a group homomorphism then $n=\operatorname{rank}\operatorname{im}\phi+\operatorname{rank}\ker\phi$. Here is my attempt at a ...
3
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0answers
35 views

Verify my induction proof for Balanced Ternary expressions

In first place I apologize if you find a grammatical error, my English is not too good for now, but I'm work on it. That also goes for errors in my question (this is my first post). I encounter the ...
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2answers
16 views

Why do I need to know that rng R = A for this proof?

Let $A$ be a nonempty set. Show that if $R$ is a symmetric and transitive relation on $A$ such that $rngR = A$, then $R$ is reflexive on $A$. So I proved this by saying: For all $x,y\in A$, $(x,y)\in ...
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0answers
16 views

Proving compactness in a geometric scenario

Let $C$ be a compact subset of $R^2$. Let $D$ be the set of all pairs of points $(P,Q)$ from $C$, such that the open segment between $P$ and $Q$ is contained in $C$: $$D = \{(P,Q)|P\in C, Q\in C, ...
2
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0answers
24 views

Regarding the expectation of a function of two random variables…

I'm trying to prove the following: If $X,Y$ have discrete p.m.f $p(x,y)$, then $\forall$ real-valued function $g$, $$E[g(X,Y)]=\sum_{x}\sum_{y}g(x,y)p(x,y))$$ I wasn't sure if my argument was ...
5
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2answers
56 views

How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
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1answer
27 views

Proof on infimum and supremum

Sorry for the poor photo quality. Can someone please tell me does my proof of this question valid or not?
0
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1answer
45 views

$f(A\cap B)=f(A)\cap f(B)$. Where's the mistake?

I'm trying to prove something that is false, to see where is the contradiction. I want to prove that if $f:X\longrightarrow Y$ and $A,B\subseteq X$ then $f(A\cap B)=f(A)\cap f(B)$. So, let $y \in ...
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4answers
25 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
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2answers
137 views

Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
5
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4answers
131 views

Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.

This is Atiyah-Macdonald Exercise 2.2 Exercise: Let $A$ be a ring, $\mathfrak a$ an ideal, $M$ an $A$-module. Show that $(A/\mathfrak a) \otimes_A M$ is isomorphic to $M/\mathfrak aM$. [Tensor the ...
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2answers
67 views

How can I prove this using number theory only

So this book I'm reading has this question: show that if $(a,n)=(b,n)=1$ the the equation $$ax+by\equiv c(mod( n))$$ has exactly $n$ different solutions. I was only able to prove it using ...
0
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2answers
17 views

Proving using the definitions of “strictly dominated by” and “dominated by”

Let $A, B,$ and $C$ be sets. If $A$ is strictly dominated by $B$ and $B$ is dominated by $C$, then $A$ is strictly dominated by $C$. I need to prove this using the definitions of "dominated by" ...
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0answers
51 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
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0answers
23 views

Understanding proof about $∀ j ∈ I (A_j ⊆ A _i )$

This is one of the problem which has been solved in Vellmena's How to prove book: Suppose $\{A_i | i \in I \}$ is a family of sets. Prove that if $P (∪_{i \in I} A_ i ) ⊆ ∪ _{i \in I} P (A_i )$, ...
1
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1answer
41 views

Using rank-nullity theorem to prove surjectivity and a question on injectivity

For injectivity, the proof starts with suppose $[v]_B = 0$, I'm not sure how this shows. I would think that simply saying, suppose $[v]_B = [w]_B$ and showing $v = w$ is the way to go and that comes ...
3
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1answer
51 views

Prove property of symmetric difference: if $A \triangle B \subseteq A$ then $B \subseteq A$

This problem is from Velleman p143 5. Recall from Section 1.4 that the symmetric difference of two sets A and B is the set $ A \triangle B = (A \setminus B) \cup (B \setminus A) = ( A \cup B) ...
3
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3answers
58 views

Given positive numbers $a, b, c, x, y, z$, such that $a + x = b + y = c + z = S$, prove that $ay + bz +cx < S^2$

Given positive numbers $a, b, c, x, y, z$, such that $a + x = b + y = c + z = S$, prove that $ay + bz +cx < S^2$ One solution is: Denote $T = S/2$. One of the triples $(a, b, c)$ and $(x, y, z)$ ...
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0answers
40 views

Proof for solving linear congruence

Show that the equation $$ a x \equiv b \bmod{n} \; \quad a, b \in \mathbb{Z}\, , n \in \mathbb{N} $$ with $d:= \gcd(a,n)$ has no solution if $d \nmid b$. But if $d \mid b$, it is equivalent to $$ ...
3
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3answers
244 views

Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$

My knowledge of the fixed points and iteration equals zero, same for the notation and terminology but I really need to know if this deduction has trivial errors or is really as nice as it seems. I ...
2
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0answers
55 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq ...
2
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2answers
72 views

working backwards from $\pi r^2$

I have been dipping my toes into a bit of calculus (through the better explained website), however I have become stuck on my understanding of the area of a circle. I understand that the formula for ...
2
votes
1answer
328 views

Proof of Andrica when Assuming Oppermann

Proof of Andrica's conjecture by assuming Oppermann's conjecture. Oppermann's conjecture: $$n\geq2\wedge\pi\left(n^{2}-n\right) < \pi\left(n^{2}\right) < \pi\left(n^{2}+n\right).$$ ...
1
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4answers
65 views

What can be said about $P (A \setminus B) \setminus (P (A) \setminus P (B))$?

This is one of the problem I have been solving in Velleman's How to prove book: Suppose A and B are sets. What can you prove about $P (A \setminus B) \setminus (P (A) \setminus P (B))$ ? Now, I ...
2
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0answers
29 views

How does one prove that a computational theorem prover is correct?

There are many computational theorem provers, such as Z3 (http://z3.codeplex.com/). Such provers employ many thousands of lines of code. How can one prove that the results are correct and can be ...
1
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1answer
42 views

Proof-verification group order 48 not simple

Is the following proof correct? Let $G$ be a group of order 48. Let's prove it is not simple. $\lvert G\rvert=48=2^4\cdot3$. By Sylow's Theorem, $n_3\in\{1,4,16\},\:n_2\in\{1,3\}$. $n_3=1$ or ...
1
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1answer
23 views

Divisibility lemma: $\exists n_0\mid n,\,\, m_0\mid m,\,(n_0,m_0) = 1,\text{ and }\,[n_0,m_0] = [n,m]$

I want to prove that, in a commutative group, there always exists an element whose order is $\mathrm{lcm}$ of the orders of two other elements. The exercise indicates that it follows easily from the ...
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0answers
37 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
1
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1answer
25 views

Cantor-Bernstein theorem proof - Hrbacek and Jech textbook

I have worked through a proof of the Cantor-Bernstein theorem as presented in Hrbacek and Jech's Introduction to Set Theory third edition. It makes use of a Lemma: If $A_1 \subseteq B \subseteq A$ ...
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2answers
30 views

How many ordered bases can I find?

I have found a basis for this question, however I am curious how many correct solutions I could find? Can you explain how I would calculate this?
1
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1answer
61 views

Finding the supremum of the set $X = \bigl\{ \frac{n-1}{n} : n \in \mathbb{N} \bigr \}$

I would just like it if someone can confirm if my reasoning is correct. Firstly, I proposed that the set had an upper bound of 1 and proved it was true by induction. Then, I supposed that this was not ...
0
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0answers
26 views

Is the proof rigorous enough?

Proposition Let $n$ be a Natural Number and let $P(n)$ be a property pertaining to the Natural Numbers such that whenever $P(m{++})$ is true, $P(m)$ is true. Suppose that $P(n)$ is true. ...
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0answers
37 views

Equivalent definitions of homogeneous ideal

I need to show that an ideal $I$ of a $\mathbb{Z}$-graded ring R is homogeneous iff for every element $f \in I$, all homogeneous components of $f$ are in $I$. $\Leftarrow$ implication is obviously. ...
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0answers
32 views

Check whether the given function is differentiable at $(0,0)$

For the function$$f(x,y)= \begin{cases} \frac{x^2 \sin y}{x^2+y^2} & \text{if}\, (x,y) \neq (0,0)\\ 0 & \text{if}\,(x,y) = (0,0) \end{cases}$$ Show that ...
2
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2answers
61 views

How to identify an error in a proof?

Right now I'm studying how to find errors in proofs by looking for common mistakes such as circular reasoning, using examples etc. I haven't had too many problems for the most part but I've run into a ...
1
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1answer
32 views

local PID that is not a field is a DVR

I would be very happy if someone would check my proof of the fact that a local PID that is not a field is a DVR: Let $A$ be a local PID that is not a field. Since irreducibles generate maximal ideals ...
1
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1answer
29 views

Obtaining a Transformed Matrix

I have a matrix $$m = \begin{bmatrix} 0 & 2 & 1 & 4 & 3 \\ 1 & 0 & 3 & 2 & 4 \\ 3 & 1 & 0 & 2 & 4 \\ 4 & 3 & 1 & 0 & 2 \\ 4 & 3 ...
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1answer
61 views

Question on exercise of ideal of a point

The question was to find the ideal of a point $(\sqrt{2},\sqrt{3})$ in $\mathbb{Q}[X,Y]$ and its conjugates in $\mathbb{C}^2$. Is is correct to say that the ideal of a point is ...
2
votes
1answer
43 views

Continuous functions and existence of a root

Let $\, f:[1,2] \rightarrow \mathbb R$ be a continuous function such that for every $n$ $\in$ $\mathbb N, \exists$ $x \in [1,2]$ with $\ |f(x)| < \frac 1n$ Show that $ \exists \;c \in [1,2]$ such ...
2
votes
1answer
49 views

$f$ is derivative in $(0,2)$ that appiles $\lim_{x \to 0+}f(x)=\lim_{x \to 2-}f(x)= +\infty$ Proof there is $c \in (0,2)$ such $f'(c)=0$.

I have this problem : $f$ is derivative in $(0,2)$ that appiles $$\lim_{x \to 0+}f(x)=\lim_{x \to 2-}f(x)= +\infty$$ Proof there is $c \in (0,2)$ such $f'(c)=0$. I'll write the proof and I'll be ...
0
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0answers
37 views

Conjugation of quaternions

This proof is extreeeeemely boring, but still I must get it right. Let $x = x_0 + x_1 i + x_2 j + x_3 k \in \mathbb{H}$ (the Hamilton quaternions). Conjugation is defined as: $$x^\ast = x_0 - x_1 i - ...
2
votes
1answer
63 views

uniform continuity and equivalent sequences

Let $X$ be a subset of $\mathbb{R}$, and let $f : X\to \mathbb{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$. (b) ...