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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.

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Definition of Variable

In mathematics, what is a variable? I have encountered previous definitions stating that it is a symbol representing an element of a set or a placeholder for an unknown value. However, I have been unable to understand these definitions. Please…
B. Han
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What does the notation 'star with some group beneath it' means?

My thesis promotor sent me this text. It talks about automorphism of free groups. On the first page, the following is written: Let $\operatorname{Aut}(F_2) \to \operatorname{GL}_2(Z)$ be the homomorphism induced by the abelianization of $F_2$ (the…
Student
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What is a "rational function of $n$ variables"?

The only definition of a rational function I was able to find is that of Varsity Tutors. "A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least $1$" If we are talking merely about $x$, then I…
Sam
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How is this defined?

Just a short question: How is the set $\mathbb{Z}[\sqrt5]$ defined ? I thought this is the same as $(\sqrt5)\mathbb{Z}$, but that doesn't make sense.
Myrkuls JayKay
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Is the definition correct?

I'm writing a B.Sc. thesis in computer science and needed to include the definition of Cartesian product. I wrote it according to the following and wonder if it is correct: Cartesian product - The set of all possible permutations of two other …
Niklas Rosencrantz
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Question about definition of regular function given in Hartshorne Algebraic Geometry

I am currently working through the section on morphisms in Hartshorne's Algebraic Geometry. He introduces the notion of a function regular at a point $P$ as follows: A function $f:Y \to k$ (where $Y$ is a quasi-affine variety in $\mathbf{A}^n$, the…
Student
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What means "under scrutiny" in mathematics

What means the term "under scrutiny" in mathematics? I am not a native speaker but I understand that that "under scrutiny" means something like "being watched closely" in general meaning. As far as I understand "under scrutiny" in mathematics and…
Carol Eisen
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What is a standard curve?

I know you may find definitions out there for standard curve. But, how can we define it in a way that makes it more understandable and clear of what it does?
Simplicity
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Acummulation point

Which one of these points is accumulation point, which not and why? I read the definition x-times but I'm quite confused :-/ I also found this post which is relevant to my question but it seems to me strange. The problem is with natural numbers and…
user1097772
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Is this a valid definition for vertical asymptote?

$x=a$ is a vertical asymptote of the graph of a function $f$ if for every $\mu \gt 0$ there exists $\delta \gt 0$ such that for all $x$ where $|x-a| \lt \delta$, this implies $|f(x)| \gt \mu$.
Chad
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Why $H\neq \emptyset$ in Definition of Subgroup?

The standard definition of a Subgroup $H$ of a Group $(G,+)$ is as follows: $H$ is Subgroup of $(G,+)$ if $\begin{cases} G \supseteq H\neq \emptyset \\ \forall x,y \in H:(x+y) \in H \\ \forall x \in H:(-x) \in H \end{cases}$ Why $H\neq…
Marios
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What is the name of the rest of the triangle inequality

The trianlge inequality states (for norms) that $$ ||a|| + ||b|| \ge ||a + b|| . $$ This can also be stated in terms of the quantity $$ r \triangleq ||a|| + ||b|| - ||a + b|| \ge 0 . $$ My question is the following: is there a standard name for…
politopo
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The sum of the first n odd natural numbers

The sum of the first n odd natural numbers: $1+3+5+7+...+(2n-1)$. My question is: Why do we write the last element as $2n-1$ ? Why don't we write $1+3+5+7+...+(2n+1)$ ?
user295645
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What does "$f: [a,b] \to \mathbb R$ on a compact interval" means?

In some theorems, I see "$f: [a,b] \to \mathbb R$ on a compact interval". $[a,b]$ is actually compact, does this kind of emphasize have another meaning? Edit for a comment: "A continuous function $f: [a,b] \to \mathbb R$ on a compact interval is…
student forever
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What does [ ] define in math?

My teacher comes up with an expression saying: an + (n+1)^{2} = 1/6[n(n + 1)(2n + 1) + 6(n + 1)^{2}] I read on the internet that [ ] is used for a variaty of things. namingsly intervals, floor, etc. But in this situation what would they mean? P.S…
Nulle
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