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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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Formal way for a recursive definition

How to put this recursive definiton: Definition. (Gentzen) subformulas of A are defined by (a) A is a subformula of A; (b) if B ◦ C is a subformula of A then so are B, C, for ◦ = →,∧,∨; (c) if ∀xB or ∃xB is a subformula of A, then so is B[x := t],…
asv
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Sets and Sequences

My question is basically: true or false? I'm told this is false, but don't understand why. Set: A set is an unordered group of related elements which are distinct. Sequence: A sequence is a list of related elements which occur in a particular…
gattsbr
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Is there a modern mathematical canon

Can somebody explain me how modern mathematic knowledge is "managed". I mean: Is there an international mathematical "canon", that says what parts a valid sections of mathematics, which says that for instance, the pythagorean theorem is part of that…
Wolfgang Adamec
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Order of muliplication with or without times sign

Is 6:2(2+1)=1 or is 6:2(2+1)=9 ? I can't find any information regarding the order of multiplication without a sign. I'm sure that 6:2*(2+1) is 9, but I'm not sure about this.
Illidor
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Why does 24 not divide 12?

If t|s just means that s = tu, for some integers t and u, then why can we not say that 24|12, since 12 = 24(1/2)? Is there some additional part to the definition I am missing?
IgnorantCuriosity
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How to capture the definition of a topological space as the fixed point of a function?

There's an asymmetry between the requirements for intersections and unions in the laws for topological spaces. In the most common presentation of a topological space that I've seen, the space is a pair $(X, \tau)$ with $X$ being the "universe of…
Greg Nisbet
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why are definitions for (riemann) integration given this way?

I've noticed some authors give the definition for Riemann integration in terms of a function $f$ with domain $[a,b]$. A definition might read "A function $f:[a,b]\to\mathbb{R}$ is Riemann integrable if ...". But don't we just want $f$ to be defined…
isthisreallife
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Set-theoretic definitions of sum and product with scalar of linear maps

I use these definition, but I need set-theoretic definitions because I define $h: A\to B \text{ for }\left\{\begin{array}{l} h \subseteq (A \times B) \\ \forall x,y,z:(((x,y) \in h \wedge (x,z) \in h) \to y=z) \\ \operatorname{dom}(h)=A \\…
Marios
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For a negative series that is becoming more positive, is this an increase or decrease in magntiude?

There is a question in my textbook that asks if the magnitude of the Gibbs free energy (far right column) increases or decreases as the compound becomes longer (i.e. as you go from the top of the table to the bottom). Does magnitude mean absolute…
K-Feldspar
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what is the mathematical term for what laymen would call "space" or "spatial"?

When laymen (including most physicists) use the word "space" they generally refer to the kind of space we live in, through which objects can move, etc. This space can be non-Euclidian and higher-dimensional, in which case people would generally…
user56834
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Definition of $f$ approaches limit algebraically

I read a formulation like: "The function $f(x)$ approaches its limit value of $2$ algebraically." So I'm wondering what's the mathematical equivalant definition of "approching a limit algebraically". In other words, what do I have to show to prove…
MeLoco
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Clarification about the definition of surjectivity

Related to the question Proof that Laplacian is surjective $\mathcal{P}^n\to\mathcal{P}^{n-2}$, I know in general that the surjectivity is defined to be : $\forall f \in \mathcal{P}^{n-2}$, $\exists \hat{f} \in \mathcal{P}^n$ such that $\Delta…
user352626
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What is the official definition of $S^1$

I need an official definition of $S^1$ that is better than $\{circle\}$. The reason is because I am interesting in defining a function $f: \mathbb{R} \to S^1$ where $\mathbb{R}$ is the interval $[0, 2\pi)$, but I do not know what the image is I…
Shamisen Expert
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Definition of conjugate transpose in this case

Would somebody mind clarifying the following for me please? Suppose $\psi(f,g):=\int_a^b f(t)\overline{g(t)}dt$ where $f,g$ are complex functions of $t$, what does it mean to say that it is hermitian? How is the conjugate transpose defined in this…
harry
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What is the meaning of 'que' in math?

As part of a lengthy mathematical proof on density functions, part of the text says: We know that given $\{x_n\}_{n\in N} \subset R $ such that que $ x_n\to z\in R^c$ , we have that the corresponding times $t(x_n)\to +\infty$, so $h_1(x_n)\to…
Mehdi Haghgoo
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