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

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0answers
40 views

What is the “minimal” structure in which points and lines are defined?

Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective ...
0
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2answers
56 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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1answer
36 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
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2answers
224 views

Definition of a Minimal Set

A few times while studying math I have encountered the notion of a "minimal set". For example, given some set of subsets, what is the "minimal" sigma algebra generated by that set of subsets? Or, in ...
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1answer
22 views

interpreting the curve of intersection

I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of ...
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2answers
69 views

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
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4answers
86 views

Find the derivative of $f(x) = \sec(x)$ without the quotient rule [closed]

Part of an analysis assignment I have: " Given $f(x) = \sec(x)$, compute the derivative of $f(x)$ by using the definition of derivative. (Note that $\sec(x) = 1/\cos(x)$ and $(\cos(x))' = ...
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2answers
45 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
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1answer
22 views

Definition for trigonometric function with a different “system”

For fun, I decided to create a sort of "intuitive" (for me, anyhow) approach to degrees and such. As I can recall, degrees are based on (the Mesopotamians?)'s base $60$ math. I've read that radians ...
3
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8answers
142 views

How can expressions like $x^2+y^2 = 4$ be defined?

I'm wondering how to define the expression(?) $x^2+y^2 = 4$, because I realised it's not a function because it cannot be expressed in terms of $x$ or $y$ alone. Is it even called an expression? Of ...
3
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1answer
30 views

How do I define a block of a $(0,1)$-matrix as one that has no proper sub-blocks?

I'm struggling to come up with a definition of a "block" in a $(0,1)$-matrix $M$ such that we can decompose $M$ into blocks, but the blocks themselves don't further decompose. This is what I've got ...
3
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1answer
62 views

Extending the Ordinals analogously to the Integers

I have been (recreationally) trying to expand the notion of ordinal numbers in the same way that the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$. My objective is to be able ...
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3answers
38 views

Does Stars and Bars or the binomial coefficient represent binary sequences?

Does Stars and Bars or the binomial coefficient represent binary sequences? With the binomial coefficient we can calculate all the paths on a grid with moving up or right, that's like defining up to ...
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1answer
31 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
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1answer
57 views

Cotangent space explicit definition

Given a tangent space $T_xM$, where $M$ is a differentiable manifold homeomorphic to $\mathbb{R}^n$, we have the cotangent space $T^{*}_xM$ defined as being the set of linear functionals $\eta: T_xM ...
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0answers
19 views

Definition of “axes”

I'm working on a question which goes like this: "23. A variable rectangle is inscribed in a circle of radius 13 cm. At a certain instant one side is 10 cm long and is increasing at the rate of 3 cm ...
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1answer
49 views

What is a “right” automorphism?

Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads: If an automorphism of ...
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1answer
27 views

A kind of planar figures

Studying issues related to the planar shapes I've found some attribute, useful for my investigations: Any segment with origin in mass center and end point on figure's boundary is contained within ...
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0answers
44 views

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 ...
0
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1answer
33 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
61 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
17 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
67 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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0answers
22 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
44 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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0answers
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
51 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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0answers
57 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
34 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
86 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 $\delta = \frac 1 \varepsilon$, ...
3
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2answers
314 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
38 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
94 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
36 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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0answers
48 views

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
19 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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0answers
25 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
69 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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0answers
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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0answers
18 views

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
109 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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0answers
19 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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2answers
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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2answers
83 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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0answers
26 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
120 views

Zero: What is it and can I define it?

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 with my ignorance, ...
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0answers
64 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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0answers
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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2answers
74 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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2answers
43 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 ...