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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