For questions about the formulation of a proof, not about the mathematics behind it.

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1answer
34 views

Proof that the first pivots in matrices $A, B$ will be in the same column.

This is a part of the proof that the reduced row echelon form of a matrix $M$ is unique, so please consider that when answering this question. Now what I need to prove is this: Suppose matrices ...
0
votes
1answer
12 views

Question about invariants.

There is a list of $n$ numbers. We pick any two numbers, $u$ and $v$ and replace them by $uv + u + v$. Does the final answer after $n-1$ operations, depend on the initial choice. I noticed that if ...
0
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0answers
15 views

Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $

I am trying to formally prove: $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $ where n is an integer, and x and y are natural numbers. It is obvious that, when $\frac xy$ is ...
4
votes
3answers
164 views

Steps to prove or disprove if two rings are isomorphic

So i'm struggling on how to prove if two rings are not isomorphic to one another. My professor told me that if a ring is not isomorphic to another, the best way to prove that this is true is to find a ...
1
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3answers
44 views

Question about a specific part of proving $\sqrt 7$ is irrational

I have a question that wants me to prove that the square root of $7$ is irrational. So I know we need to use proof by contradiction, then $7 = \frac{a^2}{b^2}$ where $a$ and $b$ are coprime. Then $a^2 ...
0
votes
1answer
34 views

Need help with proofs using axioms only [duplicate]

Prove if $p,q∈R$ and $pq>0$ then either $p>0$ and $q>0$, or, $p<0$ and $q<0$ using only the field axioms. I have no idea how to do this using only the field axioms. Seems pretty ...
7
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5answers
656 views

Is this direct proof of an inequality wrong?

My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
0
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1answer
22 views

Big Omega and Not Big Omega proofs

I need to proove these three sentences: $g(n) = n + 2n^3-3n^4+4n^5$ $g(n) = \Omega(n^5) $ $g(n) \neq \Theta(5n^6)$ $g(n) = \Omega(nlogn)$ Now, for the Big Omega I have no clue how to do it, for ...
0
votes
0answers
21 views

Prove by induction that $\Gamma \vdash \varphi \Rightarrow \Gamma[x/c] \vdash \varphi[x/c]$.

As the title says: I want to prove by induction that $\Gamma \vdash \varphi \Rightarrow \Gamma[x/c] \vdash \varphi[x/c]$. I’m struggling with how to write this proof. I think I need to do induction ...
1
vote
1answer
21 views

Showing that $\lambda_n \vdash \sigma \Leftrightarrow \sigma$ holds in all models with at least $n$ elements.

As the title says, I’m trying to show that $\lambda_n \vdash \sigma \Leftrightarrow \sigma$ holds in all models with at least $n$ elements. It’s from Logic and Structure, van Dalen (2013 edition). ...
0
votes
2answers
35 views

Strong Induction Explanation

I would like an explanation of the principle of strong induction in general, as well as a formal statement of how to prove a statement true for some subset of integers using it. Specifcally, I am ...
0
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0answers
9 views

Conceptual Question on Takens embedding Theorum

I am from signal processing background and so unaware of many details of Takens phase space reconstruction theorum. Reading the paper : A First Analysis of the Stability of Takens’ Embedding download ...
1
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1answer
30 views

Proving $(A-B) \times C = (A \times C) - (B \times C)$

For all sets $A$, $B$, and $C$, $(A-B) \times C = (A \times C) - (B \times C)$. Does this work right to left if you're assuming $y$ is both $\in$ and $\notin C$?
0
votes
1answer
26 views

How to disprove the following using negation?

Let $\mathcal{F}=\{f|f:\mathbb{N}\to\mathbb{R}^+\}$ Disprove $\forall f,g\in\mathcal{F}:$$\log{f(n)} \in O(g(n)) \implies f(n) \in O(3^{g(n)}).$ (Here we assume log has base 2) (We disprove) Let ...
0
votes
1answer
14 views

Proof of sample mean and median

Show that $$\overline{X}=arg_{a\in\mathbb{R}}min \sum_{i=1}^n (x_i-a)^2$$ $$m=arg_{a\in\mathbb{R}}min \sum_{i=1}^n |x_i-a|$$ where $m$ is the median Is there any way to prove it? I tried ...
-1
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1answer
12 views

Prove that element of the set is divisible by k

How do you prove this theorem? Theorem. For all positive integers n and k, some element of the set {n, n + 1, n + 2, . . . , n + k − 1} is divisible by k
1
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2answers
81 views

Proof using only field axioms

Prove if $x, y ∈ R$ and $xy > 0$ then either $x > 0$ and $y > 0$, or, $x < 0$ and $y < 0$ using only the field axioms. These include the Field axioms for addition, multiplication, ...
2
votes
1answer
42 views

Proofs Involving an Algorithm

Okay. I've been trying to work on a math proof and then I fell asleep. I feel as if it should be obvious but I'm not getting it at all. The following is the info: $$\frac{A}{B} = \frac{1}{n_{1}} + ...
2
votes
1answer
45 views

Show that $r$ is the rank of the $n$x$n$ matrix $A\iff A$ has a nonsingular $r$x$r$ submatrix [duplicate]

Show that $r$ is the rank of the $n$x$n$ matrix $A\iff A$ has a nonsingular $r$x$r$ submatrix, but any larger square submatrix of $A$ is singular. I know that to be nonsingular, det $ \neq 0$ I ...
0
votes
0answers
20 views

Proof strategy - How to prove this modeling of time series

The question is based on a paper titled : Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks On page 2 right above Eq(1), the authors say ...
0
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0answers
35 views

Prove that any matrix can be reduced to echelon form by an appropriate sequence of elementary row operations.

Prove that any matrix can be reduced to echelon form by an appropriate sequence of elementary row operations. I know the elementary row operations are: $E_1$:Interchange the $i^{th}$ row and ...
0
votes
0answers
33 views

How to prove or disprove $\forall f\in\mathcal{F}: \lfloor \sqrt{\lfloor f(n)\rfloor }\rfloor \in O(\sqrt{f(n)})$?

If $\mathcal{F}=\{f|f:\mathbb{N}\to\mathbb{R}^+\}$ How to prove or disprove $\forall f\in\mathcal{F}: \lfloor \sqrt{\lfloor f(n)\rfloor }\rfloor \in O(\sqrt{f(n)})$ . So I tried various functions ...
0
votes
0answers
36 views

How to prove $\forall f,g\in\mathcal{F}: \log{f(n)} \in O(g(n))\implies f(n)\in O(3^{g(n)})$ if $\mathcal{F}=\{f|f:\mathbb{N}\to\mathbb{R}^+\}$?

Let $\mathcal{F}=\{f|f:\mathbb{N}\to\mathbb{R}^+\}$ How to prove or disprove $\forall f,g\in\mathcal{F}: \log{f(n)} \in O(g(n))\implies f(n)\in O(3^{g(n)}).$ I think it can be proved. Equivalently, ...
0
votes
1answer
53 views

Discrete Measure of Interval

In Terry Tao's Measure Theory, he notes that the length of an interval $|A| = b-a$ may be recovered as \begin{equation*}|A| = \lim_{N \to \infty} \frac{1}{N} \# \left(A \cap \frac{\mathbb Z}{N} ...
0
votes
0answers
14 views

If $AX=B$ and $A'X=B'$ are equivalent if $[A|B] \sim [A'|B']$, show that inverse is not true

It is known that : if $AX = B $ and $A'X = B'$ ; then $ [A|B] \sim [A'|B']$ On the other hand, how can I show that converse of that definition is not true for all conditions.
0
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0answers
10 views

Proving weak simulation

I want to prove something but I am not sure if it is the right way to do it. I have two LTS that define different semantics. A=($Q_a,Λ,\to)$, and B=$(Q_b,Λ\cup\{\beta\},\leadsto)$, where $\beta$ is ...
1
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1answer
35 views

Inequality for Gamma function

Prove that $$0<\frac{\Gamma(x+y)}{\Gamma(xy)-1}\leq3$$ for all $x>0,y>0, xy>2.$ And equality holds $x=y=2.$
0
votes
1answer
43 views

Simple proof using only field axioms

I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both ...
0
votes
2answers
26 views

If $A⊈B∪C$ then $A-B⊈C$

This is the statement that needs to be proved: If $A⊈B∪C$ then $A-B⊈C$. I want to proof using the contrapositive, so: If $A-B⊆C$ then $A⊆B∪C.$ And I don't know what to do from this point on. ...
0
votes
1answer
82 views

Why is Mathematical Induction used to prove solvable inequalities?

As a first year undergrad student I've seen problems where solvable inequalities need to be proven to hold in a specific domain using Mathematical Induction. My question is, if the inequalities are ...
4
votes
4answers
91 views

Prove that $x^2+1$ cannot be a perfect square for any positive integer x?

I started this problem by trying proof by contradiction. I first noted that the problem stated that $x$ had to be a positive integer, and thus $x=0$ could not be a solution. I then assumed that ...
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votes
3answers
55 views

if $a,b\in\mathbb{R}$ proof $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ [closed]

$a,b\in\mathbb{R}$ ; $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ please prove this inequality system...
0
votes
1answer
53 views

(trivial proof) show that $(x-1)(x-3)\geq -2$

I'm taking my first math course that requires me to write proofs, and even though I understand most of the course material, I'm struggling with actually proving things in a rigorous way. For example, ...
0
votes
2answers
37 views

Prove the following Theorem. For all positive integers n, some element of the set {n, n + 1} is divisible by 2.

So far I got: Either n=2q or n=2q+1 where q is an element of Z. If n=2q where q is an element of Z then n is divisible by 2. I'm having trouble figuring out if that is correct and testing if n+1 ...
2
votes
1answer
48 views

Combinatorial Proof of Derangement Identity

Let $D_n$ be the number of derangements of n objects. Find a combinatorial proof of the following identity: $n! = D_n + \dbinom{n}{1} \cdot D_{n-1}+ \dbinom{n}{2} \cdot D_{n-2} + \cdots + ...
3
votes
1answer
63 views

Arzela-Ascoli Theorem on metric spaces

I've been looking for a proof of one particular direction of this theorem for metric spaces. I've looked online, but everyone seems to use different terminology/notation to state the theorem, so I'd ...
2
votes
1answer
19 views

Proof verification in a functional analysis problem

I am new to Functional Analysis .Please review the following proof: Let $X$ be a Banach space. Let $T:X\to X$ be a invertible linear operator and $M>0$ be such that $\|T^{-k}\|<M$ for all ...
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0answers
103 views

Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions.

Prove that the equation $x^4 = y^2 +z^2 +4$ has no integer solutions. I believe I have proved it for the case when both $y$ and $z$ are of the same parity. Case 1: When $y$ and $z$ are of the ...
2
votes
2answers
69 views

Problem in measure theory -proof verification

Let $f:\Bbb R\to \Bbb R$ be a bounded Lebesgue measurable function such that for all $a,b\in \Bbb R$ with $-\infty<a<b<\infty$ we have $\int _a^b f=0$. Show that $\int_Ef=0$ for each ...
0
votes
1answer
30 views

Please help how to show that $x_{n_k}$ is convergent

In a normed linear space $X$ if every absolutely convergent series is convergent then prove that the space is a Banach Space. My try: Let $x_n$ be a Cauchy Sequence in $X$ .Then we can find a ...
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votes
1answer
42 views

Proving that if a vector belongs to a null space then a multiple of it also belongs to a similar nullspace

Given that $A$ and $B$ are similar matrices with $B=P^{-1} A P$ how can I prove that if a vector $\vec{x}$ $\in$ Null(B), then P $\vec{x}$ $\in$ Null (A)
0
votes
1answer
31 views

If $f_1, f_2, f_3,\ldots$ is the Fibonacci sequence proof $f_1^2 + f_2^ 2 + \cdots + f_n^2 = f_n f_{n+1}$. [duplicate]

I'm assuming this is using strong induction/ regular induction. However, besides the "base case" I'm really confused with the inductive steps in my notes. The inductive steps in my notes use the ...
2
votes
2answers
58 views

How to prove that continuous function do not necessarily preserve cauchy sequences

I am trying to construct a proof that continuous function do not preserve Cauchy sequences Every proof I can find is disprove by counter example, which is great but these counter examples cannot be ...
0
votes
3answers
73 views

How does logic and elementary set theory work together to prove $A \cup \varnothing = A$?

In How do I prove $A \cup\varnothing = A$ and $A \cap\varnothing = \varnothing$ A proof was given reproduced here: Prove: $A \cup \varnothing = A$ Let $a\in A\cup \varnothing$. Then $a\in A$ or ...
0
votes
1answer
33 views

Correctness of Idea of Big O Proof

I have this big O proof and was wondering about the correctness of my rough work. Could anyone confirm if my idea for my proof is correct? Here is the question: Let ...
-2
votes
1answer
25 views

Proving/disproving statements with a given context of natural numbers.

How do I prove the following statements or their negations in the context where $x$ and $y$ are rational numbers in the closed interval $[-\sqrt{2}, \sqrt{2}]$? Statement 1: $\forall x \exists y\; x ...
0
votes
1answer
30 views

Proving $m^*(A \cup B) \leq m^*(A) + m^*(B)$

Let $m^*(A) = \inf\{\sum\limits_{k=1}^\infty {E_k}: \{E_k\} \text{ is a cover of } A\}$ where each $E_k$ is an interval (i.e. continuous open set) Prove: $m^*(A \cup B) \leq m^*(A) + m^*(B)$ ...
0
votes
0answers
10 views

Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
1
vote
1answer
42 views

Is it necessary to prove equality from both sides?

I have asked this question yesterday, and my friend told me, to rememeber to "prove it" also from the other side e.g. Let x $\in$ Conv($M+u$).....then $x$ $\in$ Conv($M$)+ $u$. Why would somebody ...
1
vote
0answers
37 views

Help on Big O proof

I need some help with a big O proof. I think I have a proof but I feel like some of the steps aren't logically compatible. The Question: For all functions f,g with domain $\mathbb{N}$ that maps to ...