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

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Hausdorff Distance on Fuzzy Sets

I'm trying to define the Hausdorff distance between two fuzzy sets in terms of non-fuzzy sets. Is this a viable definition? How can I show that this reduces to the Hausdorff Distance for non-fuzzy ...
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1answer
20 views

n-tuple function definition

I've read this definition for an hour now and I cannot piece it together abstractly. To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the ...
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1answer
38 views

How many teams of $5$ players out of $15$ girls and $10$ boys can be formed with at least $2$ boys and $2$ girls [with complement]

How many teams of $5$ players out of 15 girls and 10 boys can be formed with at least 2 boys and 2 girls? The solution has to be with complement. This is related to: How many ways to assemble a ...
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1answer
12 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
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1answer
63 views

What is the difference between an antiderivative and an integral?

In my textbook, it states the fundamental theorem of calculus as follows: If $f(z) $ has an antiderivative $F(z)$, then $\int^{z_2}_{z_1} f(z)dz=F(z_2)-F(z_1)$. There isn't a definition of what an ...
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17 views

Definition of multimodule

Reference : Bourbaki - Algebra I p.224 Let $\{A_i\}_{i\in I}$ and $\{B_j\}_{j\in J}$ be families of rings. Let $M$ be a set such that for each $i\in I, M$ is a left $A_i$-module and for each ...
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1answer
31 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
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36 views

Use of the word “if” in mathematical definitions [duplicate]

I'm looking at the following definition The random variables $X_{1}, \ldots, X_{d}$ are said to be comonotonic if they admit as copula the Frechet upper bound. I am however not quite sure how to ...
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1answer
47 views

Is free abelian group a “free” abelian group or “free abelian group”?

Let $G$ be an abelian group. What does it mean that $G$ is a free abelian group? Does this mean that $G$ is a free group or a free-$\mathbb{Z}$-module with the operation $n•a=a+...+a (n-times)$? Or ...
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56 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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1answer
25 views

Dirac Delta Function definition with ksi (ξ)

The dirac delta function has a definition $$f(0)=\int_∞^∞f(x)δ(x)dx$$ and $$ f(x)=\int_∞^∞f(x-ξ)δ(ξ)dξ $$ (the lower bound is minus infinity but I couldn't add a minus :/) I do understand the ...
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1answer
83 views

If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$? [closed]

For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$? If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, ...
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2answers
292 views

Why use open sets in definitions?

I've been wondering why do we use mostly open sets in mathematics to define numerous things. For example continuity is defined using (open) neighbourhoods, differentiability and Taylor's theorem is ...
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1answer
36 views

What do these definitions of conjugacy have in common?

Here are four (seemingly) different uses of the word conjugate: Complex conjugates are a concrete instance of the idea of conjugacy in field extensions. In group theory, there's the idea of ...
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2answers
74 views

Memorising lots of maths theorems/lemmas

In a few weeks I'll have my summer exams in a Senior Freshman Mathematics course. Two of my modules have a huge number of theorems, lemmas and definitions - over 200 definitions and 250 proofs by my ...
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1answer
28 views

Definition: What is a two-sided Lie ideal of a Lie algebra?

Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a subalgebra. According to wikipedia, $\mathfrak{h}$ is called an ideal of $\mathfrak{g}$ if it satisfies the condition that ...
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What is the standard definition of Torsion element?

Here are two different definitions of torsion element. Let $M$ be an $R$-module and $m\in M$. Wikipedia: $m$ is a torsion element iff there exists a nonzero regular element $r$ (i.e. Not a zero ...
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1answer
16 views

Different percent-valued Majority gates

It has been defined that a Majority gate follows this formula concerning its behavior about outputs: $$\operatorname{Maj} \left ( p_1,\dots,p_n \right ) = \left \lfloor \frac{1}{2} + ...
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24 views

Definition of a topology on a set $X$: may $I$ contain uncountable many elements or is it restricted to be finite or countable?

I've the following definition of a topology $\mathcal I$ on a set $X$: (T1) $U_a \in \mathcal I, a \in I \Rightarrow \cup_{a \in I} U_a \in \mathcal I$ (T2) $U_1, U_2 \in \mathcal I \Rightarrow U_1 ...
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1answer
48 views

About the definition of universal covering space

There are some references (for instance in Greenberg & Harper) that consider the universal covering space to be not only simply connected but also locally path connected. This definition seems to ...
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23 views

Why do we define the ordered pair in this way? [duplicate]

When we define an ordered pair (x,y) in a set, why, in many textbook, do we define it as {x,{x,y}} or {{x},{x,y}} instead of {x,{y}} or {{x},{y}} which obviously makes more intuitive sense if we ...
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What is a “nonparametric function”?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I ...
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3answers
101 views

Field axioms: Why do we have $ 1 \neq 0$?

In the definitions of a field, we have $ 1 \neq 0$. I know that in regular multiplication $0 \times 1=0$ but for reciprocal we don't have inverse of $0$. But all the spaces and different definitions ...
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15 views

definition of limit at a point and relationship with gradient

Why is $\lim\limits_{t\to\infty}\frac{f(x^*+td)-f(x^*)}{t}=\nabla f(x^*)^Td$ for $t>0$ and sufficiently small?
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58 views

Proper divisors of 1?

What are the proper divisors of 1? I understand proper divisors do not include the number itself, but is 1 an exception or does it have no proper divisors?
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76 views

Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ ...
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25 views

What the meaning 'directed'

What's the meaning of 'directed' and 'cofinal'.It's about a partially ordered set. Please give me an example? A preordered set (I, ≤) is directed if every finite subset F of I has an upper bound. A ...
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2answers
62 views

Zero: What is it and can I define it? [closed]

Struggling here... So I understand that zero is utterly and completely necessary. I've been reading a lot about this and it seems like some people get heated. I'm not a math guy, so if I offend you ...
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61 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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37 views

why does function relates input to only one output? [duplicate]

I'm trying to understand the rationale behind definition of function. Why is function defined as a relation which relates input to only 1 output? What would happen if we allow function to relate ...
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68 views

Trigonometry is to triangle as _____ is to circle.

What is the most suitable word to put in that gap? Something that corresponds to the study of circles.
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40 views

Group action and Right action

Sorry if this may seem trivial - I just started studying Group Theory. This is the problem: Prove that $(g,h) \rightarrow hg$ does not define a group action with $g$ acting on $h$. Prove instead ...
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1answer
18 views

Definition of an induced matrix norm.

Could someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in ...
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1answer
36 views

What's the difference between list and sequence (as mathematical concepts not programming point of view)?

What distinguishes between set, multiset and list is whether the order is important or not, and whether repetitions of elements is allowed: List: order is important and repetition is allowed ...
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Definition of argument-based $\min$ and $\max$

When I say "argument-based", I mean $\min$ and $\max$ functions that take arguments, as opposed to the min and max of a function. For example: $\min(3,4,-2)=-2$, $\min(100,3)=3$, $\max(3,4,-2)=4$, ...
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Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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2answers
50 views

What is $\mathbb{Z}[a]$?

Let $\alpha$ be algebraic over a field $F$. Then, $F(\alpha)$ denotes the subfield of $F$ generated by $\alpha$. This is the standard definition of $F(\alpha)$. Under this definition, for example, ...
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1answer
26 views

What is the justification for calling a hereditary system an independent system?

I was learning about set systems and hereditary systems and I noticed that they also call a hereditary system a independence system and that didn't quite make sense to me intuitively. First recall ...
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249 views

Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequence ...
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1answer
91 views

Left Hand Derivative Definition

What is the actual definition of Left Hand Derivative ? I bumbed into this site and the second white box on their site gives the definition . Is that wrong ? What is the correct one then ?
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73 views

Precise definition of distinct vectors

What is the precise definition of 'distinct vectors'? In particular, are the vectors (2, 1) and (4, 2) distinct, seeing as they are multiples of each other?
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54 views

Definition of algebraic structure

Is there a definition of algebraic structure? Wikipedia says: a set (called carrier set or underlying set) with one or more finitary operations defined on it. In particular, what is the ...
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18 views

Complex conjugate of operator on differential forms

In a Lecture note on Kahler manifolds, the author writes the following two identities are equivalent as they are conjugates of each other : $[\Lambda, \bar{\partial}]=-i\partial^{*} $ and ...
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51 views

How to say this in math lang?

Well, I have a paper to write in which I have to formalize the following definitions: A transaction is composed of actions (and cascading actions), which affect tables, which are made of records. ...
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1answer
38 views

Is it legal to define a function to be undefined for some x?

Is the following definition legal: f(x) = 1 for 0<x<1, and undefined otherwise. Such functions clearly exist, but the question is if it is legal to just ...
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Is $\int^{\infty}_{-\infty}\delta(x-x_0)f(x) \, dx = f(x_0)$ sufficient to define delta distributions?

Most of the sources start introductory section of the delta distributions by defining \begin{eqnarray} \delta(x-x_0)&=&\begin{cases} \infty, & \text{if $x=x_0$}.\\ 0, & ...
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43 views

Definition of Euclidean domain

Today we learned about Euclidean domains in class but I don't understand why we need one of the conditions stated in the definition . We called an integral domain R a Euclidean domain if there exists ...
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1answer
65 views

A strange definition for a strange set.

Given a ring $A$ we know that its center $C=\{x : yx= xy \;, \forall y \in A\}$ is a well defined subset of $A$. Now I want define a set that, intuitively, is '' The set of all elements that commute ...
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65 views

Is definable $\emptyset$-definable?

A set $A$ is $X$-definable in an $L$-structure $\mathcal{M}$ with a domain $M$ iff there are an $L$ formula $\phi(a,\bar x)$ and $\bar x \in X^n$ such that $A=\{a \in M^m | \mathcal{M} \models ...
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1answer
85 views

Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?