Mathematics Stack Exchange
Mathematics Stack Exchange

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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How to define "closer to proportion"

$\def\prop#1#2#3{#1:#2:#3}$Let's say I have a proportion, $\prop 135$ And then a set of other ones, $$ \begin{array}{l} \prop 235\\ \prop 145 \\ \prop 136 \end{array} $$ Which one is closer to $\prop 135$? How to define it closer with…
Shane Hsu
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How to interpret epsilon $(\epsilon >0)$ when used as "any" in a definition

This is something elementary, so I need to have a practical notion as well as a theoretical one for definitions and theorems that include a positive epsilon number $(\epsilon >0)$. What does this number mean, how is it interpreted? In the following…
Neo Avi
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Help in understanding a paragraph from 'What is Mathematics'?

"Whatever our philosophical standpoint may be, for all purposes of scientific observation an object exhausts itself in the totality of possible relations to the perceiving subject or instrument. Of course, mere perception does not constitute…
Ant Mani
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Is it valid to talk about a smaller amount by referring to it as a multiple less than another number?

It always feels wrong when I read something like "ABC is 3 times less likely than DEF", when it would make more sense to just say "ABC is a third as likely as DEF" or "DEF is 3 times more likely than ABC". Is the "3 times less" usage against any…
Brian Taylor
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Are supermarket pricings considered rates?

Small little disclaimer, it is my first time posting a question, so any feedbacks on what I should improve on when asking questions would also be appreciated. On to the question proper: I am a teacher and one of the next topic I am preparing for is…
Njh332
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Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.?

When we define a new mathematical structure, we generally double up on definitions. We define structures (think: metric spaces, partially ordered sets, etc.) and also the ingredients that they're built up from (think: metrics, partial orders,…
goblin GONE
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function equality

The following really gets confusing: As a simple example: Is the function $f(x) = 1$ the same function as $f(x) = \frac{x - 1}{x-1}$? $f(x) = \frac{(x-1)(x+1)}{x-1}$ the same function as $f(x)= x + 1$? Basically: does resolving and simplifying a…
MuseOfMusic
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Different definitions of the same object in math and how they are related

While reading I came across in a book with two different definitions about the same mathematical object. This kinda make feel anxious. Let me explain precisely what I mean. Cartesian product can be defined in two different but I assume equivalent…
Brian de León
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What is the difference between recursion and induction?

What is the difference between recursion and induction? I have heard those terms used interchangeably, but I was wondering if there is a difference between them, and if so, what the difference is.
user107952
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Deterministic function

In this paper the author speaks of "deterministic function". I could find no standard definition of what a deterministic function is. Do you know what he might be referring to?
Cesare
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What does it mean by the term "x-polynomial"?

What does it mean by the term "x-polynomial"? Let say I have a problem like this: Consider the polynomial $2x^2-5xy-3y^2+x+11y-6$ as $x – polynomial$. Find the degree and numerical coefficient of each term. In my understanding, this means that all…
AYA
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how does a surjective function not contradict the definition of a function.

I was reading up on functions and different types of functions. From wikipedia : A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y Surjective…
L.Emil
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What does the notation $d\langle W_1, W_2\rangle$ mean?

Let $W_1(t)$ and $W_2(t)$ be standard Brownian motions for $t\geq 0$. I have come across the notation $$d\langle W_1, W_2\rangle$$ in a text I am reading, but I cannot find the definition for it. I assume it a standard term in the study of…
Xiaomi
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Fourier transform

Could anyone explain to me how do we change Fourier transform equation from this [Wiki - look at the top of the page]: $$ \mathcal{F}(x) = \int\limits_{-\infty}^{\infty} \mathcal{G}(k)\, e^{-2\pi i k x} \, \mathrm{d} k $$ to this [Wiki - check…
71GA
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Do "embedding" and "injective homomorphism" mean the same thing?

On page 6 of A Shorter Model Theory, it says For example if $G$ and $H$ are groups and $f : G \to H$ is a homomorphism, then (2.1) says that $f(1^G)=1^H$ and $f(a^{(-1)^{G}}) = > f(a)^{(-1)^{H}}$. This is exactly the usual definition of…
Greg Nisbet
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