Questions tagged [proof-writing]
For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.
16,100 questions
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Any easier ways to prove an explicit form for a generated $\sigma$-algebra besides transfinite induction?
This question is a follow-up to an answer to a previous question, and motivated by my laziness in not wanting to learn about transfinite induction or how to write proofs using transfinite induction ...
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Can you prove equality of two expressions by setting them equal in an equation? [closed]
Suppose I have two expressions, and I wish to prove that they are equal to each other. Must I perform algebraic operations on one of the expressions in an attempt to reach the other one? Or perhaps ...
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The exact meaning of ‘subject to that’ in this context
In the following sentence from the paper (see Page 4, the proof of Lemma 3.4) (see the paper in https://doi.org/10.1016/j.disc.2023.113431) on extremal graphs:
Let $G$ be an edge-extremal graph in ...
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Upper-bounded version of the Gale-Ryser theorem
The standard Gale-Ryser theorem is for the existence of a $(0,1)$-matrix given exact row sums $R = (r_1, \dots, r_K)$ and exact column sums $C = (c_1, \dots, c_M)$. What if we relax the column sums ...
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Motivation for constructing auxiliary functions in a proof that $f(x) \to0$ given a differential inequality
I'm working on a problem in analysis and I understand the steps of the proof for one of its cases, but I'm struggling to understand the motivation behind the specific construction used. I'd appreciate ...
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"where" in math text [closed]
I read somewhere that using the word "where" in the text immediately after an equation is not good style in math prose. Instead, the statement you might make after the word "where" ...
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Proof writing standard: English vs Symbols. What's better? [closed]
I’m new to proof writing. For a general proof, I’ve come across books writing proofs by use of formal grammar and math. Take this common textbook example,
(1) Proposition: If $x$ is even, then $x^2$ ...
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Is an "algebraic proof" considered to be its own category type of proof?
If we have a proof for the derivation of a formula, which primarily relies on substituting terms with equivalent terms and simplifying them (i.e. combining like terms and using the addition, ...
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Uniform Continuity Real Analysis Proof
Let $a < b < c < d$ be real numbers, and suppose $f : (a, d) \to \Bbb{R}$ is a function such that $f$ is uniformly continuous on $(a, c)$ and also uniformly continuous on $(b, d)$. Prove that ...
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What are the dihedral angles that an inscribed tetrahedron makes with its cube?
Inscribe a regular tetrahedron in a cube. What dihedral angles do its faces make with the faces of the cube?
Proposed Solution: The angles formed fall into two categories:
Where their intersection ...
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How to develop a cone into a sector using synthetic geometry only?
Consider a right circular cone with radius $r$ and slant height $s$. Its surface area is
$$
A = \pi r s.
$$
Proof: It suffices to show that the cone can be sliced and unwrapped, without deformation, ...
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Demonstration of $ \left( 1+\varepsilon \right)^{x}\ge1+x\varepsilon $
consulting:
Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
Simil Bernoulli inequality for induction
I follow a proccedure on we Let $ \varepsilon\gt-1 $ and $ x $ a positive ...
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How to concisely assert the existence of something in an assumption?
One might state the following:
Let $G$ be a group and $p$ a prime such that a power of $p$ divides $|G|$. By the Sylow theorems, let $S$ be a Sylow $p$-subgroup.
This statement assumes $S$ is a ...
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The number of divisors of $n$ and the sum of divisors of $n^2$?
Let $\tau(n)$ be the number of positive divisors of $n$, and let $\sigma(n)$ be the sum of its positive divisors.
I was playing around with these functions for small values of $n$ and noticed ...
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What Makes a Proof (and How to Write One)?
While I'm aware this is unlikely to be an uncommon question, I am going to ask it within the context of my explorations in mathematics.
I am taking a crack at Michael Spivak's Calculus (3e), along ...
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How should I write proofs requiring complicated, yet routine, symbolic manipulation?
Often proofs may involve multiple lines of routine symbolic manipulation (e.g. taking derivatives, applying routine identities, or routine algebraic manipulations) which are distracting + tedious.
How ...
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Basis of a topology - Equivalent definitions confusion
I know that similar questions were already asked here, here and here. Unfortunately I wasn't able to understand the equivalence of definitions from any of answers (to any of the linked questions). I ...
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How to show that if $(X_i)_{1 \leq i \leq n}$ are an ordered n-tuple of sets, then the Cartesian product, is a set
I am trying to do the following exercise from Tao's Analysis I but I think I may not have the correct intuition on how to approach the proof:
Show that if $(X_i)_{1 \leq i \leq n}$ are an ordered n-...
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Proving that every power of 2 is slightly deficient
I'm working on a number theory problem about deficient numbers and would appreciate some guidance.
The exercise is:
Prove that every power of 2 is slightly deficient.
I understand that a number $n$ ...
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What is the difference between these two groups, and which is correct when writing the proof? [closed]
let :
\begin{align*}
G &= \{e^{\frac{i\pi k}{3}} , k \in \{0 , 1 ,2 ,3,4,5\} \}\\ G_1 &= \{e^{\frac{i\pi k}{3}} , k \in \mathbb{Z}\}
\end{align*}
we have :
\begin{align*}
G_1 &= \{ e^{\...
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Recommended book or note for methods of mathematical proof [duplicate]
I asking about good book for undergraduate student to learn methods of mathematical proofs in more details and has lot of examples. I found "book of proof by richard hammack" but I want more....
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Proof of compactness of the idèle class group of norm one using finiteness of class group and Dirichlet unit theorem.
$\DeclareMathOperator{\Pic}{Pic}$
This is a question about a proof from Neukirch's Algebraic Number Theory (Chapter VI, Theorem 1.6, page 362), and I mostly use his notation.
Let $I_K$ be the idèle ...
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3 variables cyclic rearrangement inequality.
For positive $a, b, c$, prove that $$\sum_{\text{cyc}}\frac{a^2b(b-c)}{a+b} \geq 0.$$
The LHS being symmetric, we can assume $a \ge b \ge c$. I tried combining the denominators ; since $(a+b)(b+c)(c+...
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Is there a standard classification of mathematical proof methods? [closed]
It seems it is generally known what a mathematical proof is, and why they are important. For example, What is a proof?, What are some common proof strategies in
mathematics?, Can a proof be just words?...
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4
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How does the ϵ-δ definition inherently ensure that ϵ decreases as δ decreases for non-constant functions?
When we intuitively describe the limit of a function, say f(x) at point a equals L, we mean that as x gets closer to a, then f(x) gets closer to L. This is the intuitive concept of a limit. ...
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Help to Justify the Proof: If a Curve Lies on a Line, Then Its Curvature Is 0
I don't know how to write the proof properly. The statement is: Let $\gamma : I \subset \mathbb{R} \rightarrow \mathbb{R}$ be parametrized by arc length such that it lies in a straight line, then $k(s)...
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Potential proof of Hartshorne Exercise I.4.7.
I've been trying to prove this exercise for a while, and I've finally got something that vaguely looks right. For reference, the question is:
Let $X$ and $Y$ be two varieties. Suppose there are ...
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Almost everywhere and complete measure
Definition: $A$ measure space $(X,\mathcal{M}_{cpl},\mu_{cpl})$ is called complete if $N \in \mathcal{M}_{cpl},\ \mu(N)=0,\ A \subseteq N \ \implies \ A \in \mathcal{M}_{cpl}.
$
Let $(X,\mathcal{M},\...
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Prove that every future directed causal vector is a limit of future directed timelike vectors
I want to prove the following property of vectors in Lorentzian Geometry.
Let $p \in M$ and $f: T_{p} M \rightarrow \mathbb {R}$ where $f(X)=T(p)(X,.,.,.,X)$ be multilinear evaluation at p. Assume ...
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Proof techniques to analyze a recursively defined estimator
I’m studying a recursively defined estimator and want to prove consistency. The estimator is not merely computed by a recursive algorithm—the value on a sample is defined (also) through values on sub-...
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Prove that every point of n-dimensional rectangle is inside the n-dimensional sphere.
Let $R \subset \mathbb{R}^n$ be a closed axis-aligned rectangle (i.e. a box) with vertices
$$
a_1, a_2, \dots, a_{2^n}.
$$
Define
$$
r = \max \{ \lVert a_1 \rVert, \lVert a_2 \rVert, \dots, \lVert a_{...
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Why is the sequence of maximum reachable values monotone in this jumping problem?
For context, this math question is inspired by this Leetcode problem.
Let $ \text{nums} $ be an integer array. From any index $i$, we can jump to an index $j$ under these rules:
If $j > i$, the ...
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1
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Fixing the circularity of a proof: "we may assume $\beta=\infty$"
I would like to elaborate on a key step of a proof of the following theorem, from the book "Complex made Simple" by Ullrich:
Theorem $12.4$. Suppose that $K \subset \mathbb C$ is compact ...
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Clarification on bounding $\frac{1}{|x-3|}$ in $\epsilon$-$\delta$ proof for $ \lim_{x \to 5} \frac{1}{x-3} = \frac{1}{2} $
(Please note that the below is the part of the proof where you write the scratch work that is used to find a suitable $\delta$ before writing the actual formal proof.)
Hello everyone!
I am trying to ...
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Is my first Epsilon-Delta proof for $\lim_{x\to 4} \sqrt{x} = 2$ written correctly?
Body:
Hi everyone,
I’m just starting to learn epsilon-delta proofs, and I’m not yet confident in spotting mistakes in my own work. I tried to prove that
$$
\lim_{x \to 4} \sqrt{x} = 2
$$
So, here’s my ...
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What is the preferred way to prove that $A\setminus A = \varnothing?$
As a highschool student learning elementary set theory, the way we were taught to prove $A\setminus A=\varnothing$ at least from where I am was to show that $A\setminus A \subseteq \varnothing$ and $\...
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Identity of limits of sequence of sets
Let $\{A_n\}_{n\in\mathbb{N}}$ be a sequence of sets.
Assume that for any $k\in\mathbb{N}$, we have the following decomposition:
$$\bigcup_{n=k}^\infty A_n=\bigcup_{n=k}^\infty (A_n^\complement\cap A_{...
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Prove $0.6\dot{1}\dot{2} = \frac{101}{165}$ [closed]
For those who wants a bit background, this is a proof question for international GCSE paper, for students aged 14-16.
This is what the candidate has written.
Obviously, it is NOT the usual method we ...
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Simultaneously notating set membership and inequality
What is an acceptable way of concisely notating that an element $x$ is both a member of a set $X$ and is less than some number $y$?
How about:
$$
\text{Let } x \in X < y.
$$
Or:
$$
\text{Let } ...
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1
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How to rigorously robustness to outliers and prove that Least Absolute Deviation is more robust than Ordinary Least Square?
I am an engineer who really love math, and recently watched an educational video "Fitting a line WITHOUT using least squares?" where at timestamp 7:10, the presenter demonstrates that Least ...
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Given $A∩B ⊆ C\setminus D \text{ and } x ∈ A$ prove that if $x ∈ D \text{ then } x ∉ B.$
I have just started learning about proofs. I'm using Velleman's book 'How to Prove It', and I would really appreciate it if you could say what you think about my proof for this theorem:
Theorem. ...
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A proper subalgebra of a nilpotent Lie algebra is strictly contained in its normalizer - proof attempt
Let $\mathfrak{g}$ be a finite-dimensional complex nilpotent Lie algebra. Prove that for every proper subalgebra $\mathfrak{l}\subsetneqq \mathfrak{g}$ the normalizer $N_{\mathfrak{g}}(\mathfrak{l})=\...
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A rigorous proof for the geometrical problem.
Consider $\{x_i\}_{i}$ set of distinct non-zero ($x_i \ne \bar{0}$) vectors in $\mathbb{R}^n$. Assume we have the distribution over the given vectors, i.e. $p_i > 0$, such that $\sum_i p_i = 1$. ...
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Rigorous and geometric definition of curl of a video
I have always know that the curl of a vector field $\mathbf{F}$ is given from this definition:
Let $\mathbf{F} : D \to \mathbb{R}^3$, with $D \subseteq \mathbb{R}^3$ open, be a vector field of class $...
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Why any two pair of trees in Random Forest classifier have the same correlation?
According to the book Elements of Statistical Learning, also cited in the following question:
variance-of-a-random-forest suggests that any two pair of trees $f_{i}(x),f_{j}(x)$ i.i.d have the same ...
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Number of ways to arrange r distinct objects into 2 identical circles, stirling number of first kind
I consider a general term for a partition of size $k$, $r-k$.
The number of ways is:
Choose $k$ objects for the first circle: $\binom{r}{k}$
Arrange the $k$ objects in a circle: $(k-1)!$
Arrange the ...
3
votes
1
answer
123
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Does the following fruit puzzle have a proof?
There is a grocery store that ONLY has apples and bananas.
Each day in the year, John goes to a store and predicts how many apples there will be. For example, he guesses that there were 23 apples but ...
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1
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94
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how to prove the general solution theorem for homogeneous linear recurrence relations with constant coefficients?
I studied the solution of linear recurrence relations, especially the case of homogeneous linear recurrence relations with constant coefficients, which are of the form:
$$
U_n = \sum_{k=1}^m a_k U_{n-...
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2
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Set Theory Proof Questions from Textbook "How to Prove it" by David J. Velleman [closed]
I'm having issues with two problems from chapter 3.3
Exercise #4: Suppose $A \subset P(A)$. Prove that $P(A) \subset P(P(A))$
I feel completely stuck here, I know how to explain logically the given ...
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1
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Question about equilateral triangle that packed into a square
I saw this question.
Looking at the picture in this question, I have another question. If an equilateral triangle is tightly packed into a square, and one vertex of the triangle share the vertex of ...
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