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

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3answers
96 views

Is natural numbers set $\mathbb N$ infinite set?

A set with uncountable number of elements is called an infinite set. Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set? As far i know $\Bbb N$ is "countably" ...
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0answers
81 views

Uniform Spaces: Neighborhood System [closed]

Problem Given a uniform space $\Omega$. Then it gives rise to a topology via: $$\mathcal{N}_x:=\{U[x]:U\in\mathcal{U}\}\quad(U[x]:=\{y:(x,y)\in U\})$$ (See Hausdorff's Neighborhood Systems ...
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1answer
105 views

Neighborhoods: Interior

The neighborhood filters satisfy: $$\forall N\in\mathcal{N}(x):\qquad x\in N$$ $$\forall N\in\mathcal{N}(x)\exists M_0\in\mathcal{N}(x):\qquad N\in\mathcal{N}(m)\text{ for all }m\in M_0$$ Define the ...
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2answers
93 views

Manifold Orientability Definition

In Shigeyuki Morita's Geometry of Differential Forms, orientability is defined in the following way: If we can assign an orientation to each point on a manifold $M$ in such a way that the ...
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2answers
39 views

Definition of totality in relations

I see two apparently different definitions for totality which don't seem to be equivalent. Definition 1. A relation $R \subset X \times Y$ is total if it associates to every $x \in X$ at least one $y ...
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1answer
132 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
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1answer
58 views

Degree of a map $S^0 \to S^0$

By definition the degree of a map $f: S^n \to S^n$ is $\alpha \in \mathbb Z$ such that $f_\ast(z) = \alpha z$ for $f_{\ast}:H_n(S^n) \to H_n(S^n)$. What is the definition of the degree of $f: S^0 \to ...
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1answer
34 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
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3answers
562 views

True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
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2answers
98 views

Why $1$ isn't a prime? [duplicate]

I was wondering the reason behind defining the Prime Numbers in a manner of which $1$ isn't an example. I read in Rotman's A First Course in Abstract Algebra that one reason that $1$ is not called a ...
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2answers
65 views

Singular Points on Algebraic Curves

What does it mean for a point on an algebraic curve (over any field) to be singular? Over $\mathbb{R}$ or $\mathbb{C}$ is means that all derivatives vanish, but how does this definition generalise to ...
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12answers
1k views

What is a limit?

This limit thing keeps coming up in my calculus textbook. What is it?
307
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18answers
55k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
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0answers
14 views

symbol for input domain 'range'

I have a set of problems, all with differing domain definitions i.e. $x\in[-5,5]^n$, $x\in[-100,100]^n$ etc (n being the number of dimensions to the problem). I'm writing a formula that modifies an ...
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2answers
103 views

Confusion about the definition of function

Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to ...
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3answers
149 views

What's the definition of a “local property”?

Is a property called local if and only if for every point there exists a neighbourhood for which the property is true? For example: Let $X,Y$ be topological spaces. Then $f: X \to Y$ is continuous if ...
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1answer
28 views

Does this definition of “limit point” really work

I am reading Tapp's introduction to matrix groups for undergraduates. He gives the following definition of limit point of a set: A point $p \in \mathbb R^m$ is called limit point of a subset $S ...
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2answers
229 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
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1answer
46 views

is there a discipline of mathematics that studies graphical versions of various operations?

What I am interested about is a discipline that deals with mathematical operations that can be done graphically, in this case meaning using some kind of "structures" that are manipulated to arrive at ...
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4answers
215 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
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0answers
18 views

Correct definition for convergence of a subsequence?

I only have the definition for convergence of a sequence, but can't find a definition for convergence of a subsequence. I have two guesses: For all $\epsilon > 0$, there exists an $N \in ...
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7answers
427 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
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2answers
28 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
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0answers
26 views

Definition and applications of Push Down Machines [closed]

What are push down machines? Please explain in simple and short. What are the applications of push down machines ? Is DTM and NDTM are its applications ? If not then what they are ?
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1answer
48 views

Explanation for the definition of monomials as products of products

I'm attempting to learn abstract algebra, so I've been reading these notes by John Perry. Monomials are defined (p. 23) as $$ \mathbb{M} = \{x^a : a \in \mathbb{N}\} \hspace{10mm} \text{or} ...
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1answer
36 views

Tangent Space: Identifications

Given a manifold $M$. Denote a chart by $\kappa$. Introduce the directional derivative: $$\partial:\mathbb{R}^n_a\to T_a\mathbb{R}^n:v\mapsto\partial_v\rvert_a$$ That is an isomorphism with inverse ...
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1answer
15 views

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence?

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? In some places I see they call it just inc/decreasing and some call it monotonically ...
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1answer
37 views

Does the word scalar still apply if it's not a vector?

If I take the value of something, say $50$g and I multiply that value by something else, so perhaps $50\text{g} \times 3$ what would that $3$ be called? It acts like a scalar but I'm not sure that ...
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0answers
139 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
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4answers
54 views

Question about proving $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence

Prove $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence. What I did: Suppose by contra position that $n$ tends to a finite real limit $L$, so from ...
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2answers
80 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
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0answers
11 views

Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
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0answers
27 views

Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
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1answer
50 views

Definition of division algebra

The definition on Wikipedia of a division algebra $D$ is given as: Given $a,b \in D$, $b \neq 0$ there exists a unique $c\in D$: $a = bc$ and a unique $d \in D$: $a = db$. My question(s) are: What ...
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1answer
17 views

points in general position

I'm really confused by definition of general position at wikipedia. I understand that the set of points/vectors in R^d is in general position iff every (d+1) points are not in any possible hyperplane ...
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1answer
58 views

Definition of inverse function

I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative ...
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2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
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2answers
63 views

Most general definition of homomorphism and isomorphism

What is the most general definition of homomorphism and isomorphism? It is clear what they mean when there is an algebraic structure to be preserved, but what about when there is no such structure? ...
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5answers
81 views

Confusion in the definition of set

Which of the following is the correct definition for set? Set is a well defined collection of objects. Set is a collection of well defined objects.
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1answer
153 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
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1answer
33 views

Is this how the limit of a sequence of sets is commonly defined?

I was looking at the wikipedia page for the Cantor set which defined the set using a limit. I had not previously seen a limit expression involving a sequence of sets rather than real numbers, so I got ...
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2answers
78 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
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0answers
16 views

What is a nice way to call continuants?

I'm reading this paper : http://www.numbertheory.org/pdfs/continuant.pdf and here is a definition for continuant : http://en.wikipedia.org/wiki/Continuant_(mathematics) Let $\{a_n\}$ be a ...
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2answers
33 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
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0answers
25 views

What is the domain of continued fraction?

I'm trying to formally define (generalized) continued fraction. Consider $[i;\sqrt2,i,i]$. This is not well defined since $i+\frac{1}{i}=0$. What would be a domain of continued fraction? (As a ...
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6answers
6k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
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1answer
18 views

Difference between $\Delta f$ and $\Delta f(x)$

What is the difference between $\Delta f/\Delta x$ and $\Delta f(x)/\Delta x$? Are they the same? I've been watching a lecture and the professor seems to describe the slope $$m = \lim_{x\to0} \Delta ...
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1answer
43 views

Definition of $C^1, $the vector space of continuously differentiable functions

I asked a question on clarification of the symbol $C^k$. It was confirmed to me that $C^k$ is actually a space of functions. Now my next question in the definition is on $C^0$ and $C^1$. $C^0$ is ...
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1answer
29 views

Coercivity definitions

Hi I was given the following definition of coercivity: Let $V$ be a Banach space. The first definition: $A:V \rightarrow V^{*}$ is coercive iff $\exists \zeta: \mathbb{R}^{+} \rightarrow ...
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1answer
17 views

Lower Central Series and Generators

Let $G$ be a group generated by $S=\{x_1,\cdots,x_n\}$ and let its lower central series be defined as $\Gamma_1=G$, $\Gamma_m=[\Gamma_{m-1},G]$ for $m\geq 2$. By definition $\Gamma_m$ is generated by ...