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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.

7799 questions
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Sum with more indices

I have trouble understanding this way of writing a sum. How can I interpret the sum with indices $1\leq i_1<...< i_r\leq n$ ?$$\left|\bigcup_{i=1}^nA_i\right|=\sum_{r=1}^n(-1)^{r-1}\sum_{1\le i_1<\cdots
akana
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Is a number with infinite digits within the set of natural numbers?

I have 237474 ..... that number should be found within the set of natural numbers no? because if we say that the set of natural numbers has infinite elements, therefore that number should be found in the set. if that is not the case then what set is…
Cooper
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Negation of a Relational Operator

I read in a paper a definition like this: Given $\Sigma=\{\neq,=, >,<,\leq,\geq \}$, we define a constraint $v\ op \ w$ with $v,w \in \mathbb{R}$ and $op \in \Sigma$. Moreover, we indicate by $\neg (v\ op \ w)$ the constraint in which $op$ is…
kafka
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Definition of 'product' for ordered pairs

In 'Mathematics Form and Function' ch.2, section 4, 'Integers' by Saunders Mac Lane (p.50 in the 96 edition) I came across the following definitions of sum and product for ordered pairs: (m, n) + (m', n') = (m + m', n + n') (m, n) (m', n') = (mm' +…
zhanmusi
  • 171
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What ansatz means?

I often see (in the case of ODE) : Let $\dot x=f(x)+\varepsilon g(x)$ and ODE. We make the ansatz that $x(t)=Ae^{\frac{g(x)}{\varepsilon }}.$ What does it mean ? In wikipedia is not well explained. Does it mean that "we suppose that…
John
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Equivalent definition of $\text{limsup}$

Let $\{x_n\}_{n\in\mathbb{N}}\subseteq\overline{\mathbb{R}}$ a sequence. On some texts the definition of $\text{limsup}$ is as follows: Definition 1. $$\text{limsup}x_n=\inf_{k\ge1}\bigg(\sup_{n\ge k}x_n\bigg)$$ while other texts other texts give…
Jack J.
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The sum modulo $1$ on the unit circumference

Let $X=[0,1)$. For every $x,y\in X$ we define $$ x\dot{+}y:= \begin{cases} x+y & \text{if $x+y<1$} \\ x+y-1 & \text{if $x+y\ge1$} \end{cases} $$ I do not understand the following statement: This operation can be displayed as an sum of angles…
Jack J.
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Why non-increasing is decreasing?

As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing? In fact, it raises some confusing to me. For example, the…
zdm
  • 452
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"rational points are dense on the unit ciricle".

What is meant by the phrase "rational points are dense on the unit ciricle". I know the definition of a dense set: In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs…
Hidaw
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Why do definitions need to be 'proved' to work?

I am reading Elements of set theory by Enderton. I am having a conceptual difficulty with why it seems that certain definitions almost have to be 'proved' to work? Specifically, I am reading about defining an ordered pair set such that $$\langle x,y…
masiewpao
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mathematical definition of a triangle

If mathematically a square area can be defined using the Cartesian product as $\square : \left( \xi, \eta \right) \in \left[-1,1\right] \times \left[-1,1\right]$, how can I define a triangle using the same terminology? The coordinates of the…
Alejandro Marcos Aragon
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what does it mean that something fibers?

For example, in an article I have found that "compact abelian group which fibers over the circle $S^1$ [...]" and surely I have heard that phrase before. What does it mean?
HeMan
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What is a cone in $\mathbb R^n$?

What is a cone in $\mathbb R^n$ ? By logical, I would give the definition : $$\exists !p\in C: \forall x\in C, [x,p]\subset C,$$ or equivalently $$\exists !p\in C: \forall x\in C, \forall t\in [0,1], p+tx\in C.$$ Could such a definition work ? Is…
user330587
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For $0\ne{q}\in\mathbb{Q}$, is it proper to say $q^{0}=1$ is vacuously defined for convenience?

My question is two-fold. First: is $q^{0}=0$ a result or a definition? If it is a definition, is it a vacuous definition? I'm producing a collection of "review notes" to shore up the countless holes in my mathematical skills and understanding. One…
Steven Thomas Hatton
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why is angular velocity a pseudo vector but angular frequency a scalar

I note that angular velocity is a pseudo vector as direction is only either clockwise or anticlockwise. So 'I believe ?' that the direction is given by the sign, so a positive value defines an anticlockwise rotation and negative value a clockwise…
R Mcgowan
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