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Questions tagged [definition]

For requesting, clarifying, and comparing definitions of mathematical terms.

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

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Is there a word to distinguish two types of definitions?

Some definitions in mathematics are very "explicit" or "constructive". For instance, we can define a palindrome as a word obtained by taking any word W, reversing it to get another word W', and then forming a word WW' or WaW' (where a can represent…
Barry Smith
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Definition of functional derivative

In this book, A derivative of a function is defined as follows: \begin{equation} \frac{df}{dx} = \lim_{\epsilon \to 0} \frac{f(x+\epsilon) - f(x)}{\epsilon}. \end{equation} And define a functional derivative of a functional $F[f]$ as…
Orient
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What is the structural hierarchy in mathematics?

By structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, geometry, etc. At the bottom, you have you axioms,…
user2901512
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Nothing on the web; What is a Ruffini Radical

Surprisingly, it's not clearly defined online. The first thing that comes up is Abel-Ruffini theorem, which only refers to "radicals" and not RUFFINI radicals. Ian Stewart's book has it appear out of thin air as if it's prior knowledge and common to…
Melba1993
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What are the "whole numbers"?

Just recently, I attempted to answer a question involving "whole numbers", but discovered that my long-held assumption (that they're the same as the integers), is not universal. [In fact, it seems I owe a retraction for whenever I've snickered as a…
Douglas S. Stones
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Define F : Z → Z by the rule F(n) = 2 -3n, for all integers n

I am not sure how to go about solving this problem. Can somebody tell me how to define $F : Z \to Z$ by the rule $F(n) = 2 -3n$, for all integers $n$ ? I am not sure where to even start or what is meant by the question. The assignment…
User
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Why is delta used to describe difference between two entities

I hear a lot about extracting "delta" between two properties in my current job. I come from a User Interface programming background and I do not really have much math background. I looked up delta online but did not find any definition of it meaning…
open_sourse
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Definition of "or"

A quick definition clarification: Does the set $\{(x,y):x =0 \,\,\,\,\text{or} \,\,\,\,y=1 \}$ include the element $(0,1)$? (Sorry, English is not my first language, I get confused sometimes... Also sorry that this may not be a very good math…
defor
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Verifying Difference between a variable and a parameter

I'm not quite sure of the formal difference between a variable and a parameter. The way I explain a parameter is that while its input value differs from case to case, the concept is always the same. Examples include B for base, h for height, etc. …
Nate
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How can I improve my definition?

I've been trying to write a formal definition for a $k$-involutible function in that the function has to satisfy the following properties: $k$ is a positive integer. $f \in \mathbb{R}(x)$ (as in, $f$ is a rational function. I explicitly wrote out…
Arkyter
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Formal definition of a wave

I have never seen a formal definition of a wave-like function. By wave-like function, I mean, intuitively, something similar to the sine function. The closest definition I have gotten is "finite linear combination of sine functions". But this still…
user107952
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What is the correct definition of the $clip(\cdot)$ function?

I was wondering how to properly (or correctly if only one correct definition exists) define the $clip(x,min\_value, max\_value)$ function, which clips $x$ to the range $[min\_value, max\_value]$. For convenience, take $f(\cdot) = clip(\cdot)$, $a =…
Daniel B.
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On the definition of continuous function.

Let $A$ a subset of $\mathbb{R}$, let $f\colon A\to\mathbb{R}$ and let $x_0\in\ A.$ We say that $f$ is continuous in $x_0$ if for all $\varepsilon>0$ exists $\delta >0$ such that $$x\in A, |x-x_0|<\delta\Rightarrow |f(x)-f(x_0)|<\varepsilon$$ On…
Jack J.
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What is the definition of $f$ falling slower than $g?$

The German Wikipedia is full of articles like this one saying one function falls slower than another. I wrecked google, but could not find the definition or sufficient conditions. Is it: $ \lim_\limits{x \to \alpha }\frac{f(x)}{g(x)} = 0$,…
Ludi
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How is division symbol usage currently defined?

I've been in a lengthy discussion today about how to interpret the division symbol. There seem to be two views on what it means. • Everything on the left hand symbol is divided by everything on the right hand of the symbol. or • The operator only…
Benjamin Faure
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