# Questions tagged [definition]

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

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### Introduction of function symbols

According to this link, in order to introduce a new function symbol one needs to prove the formula $\forall x_1,...,x_n\exists!y:P(y,x_1,...,x_n)$. This allows for the introduction of a new $n$-ary ...
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### two different orbit definition and their explanations

I am chemistry master student and had to interact with abstract algebra somehow. I am trying to learn concept of orbit. When I look at the book, I saw two orbit definition in two distinct but somehow ...
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### Could we define a fifth arithmetic operation on real (or complex) numbers that is independent of addition, subtraction, multiplication, and division?

The four basic arithmetic operations with real (or complex) numbers are addition, subtraction, multiplication, and division. the first two being inverse operations and the last two being inverses of ...
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### binary operations on a field [closed]

i am reading the start of a linear algebra book, and reached the part of the definition of a set. the book defines 2 binary operations $+_F$ and $\cdot_F$, my question is relating to their signs, do ...
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### Multifunction Measurability

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and $X$ a Polish Space. A multifunction (or set-valued) $F:\Omega \to 2^X$ is a map from $\Omega$ into the subsets of $X$. But when defining ...
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### Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
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### Is the definition of inductive set in Discrete Mathematics the same as that in Set Theory?

James Heine in Discrete Mathematics (second edition), chapter 3, section 3.1, page 128 defines inductive set as "objects constructed, in some way, from objects that are already in the set." ...
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### Definiiton of 'Annihilator' in the context of dual map

Could you tell me what the definiton of 'annihilator' in this context ? I know the definition of annihilator for modules over ring, but in the following context, any definition I know means nothing. ...
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### Clarification on the definition of co-functions

The definition of co-functions(Wiki) is as follows: Definition: a function $f$ is cofunction of a function $g$ if $f(A) = g(B)$ whenever $A$ and $B$ are complementary angles. I would like to confirm ...
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### Multiplying Two asymmetric Laurent series

The following result $$\left(\sum_{i=0}^{\infty} a_i x^i\right)\left(\sum_{j=0}^{\infty} b_j x^j\right) = \sum_{k=0}^{\infty}\left( \sum_{\substack{j+i=k\\ i,j \ge 0}} a_i b_j\right) x^k$$ is well ...
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### For a curve to be smooth, it is necessary that its derivative is never equal to $0$. Why? (Complex Analysis, Curves in the complex plane)

I have a question about curves in the complex plane. A parametrized curve is a function $z(t)$ which maps a closed interval $[a,b]\subset\mathbb{R}$ to the complex plane. We shall impose regularity ...
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### In what sense are preadditive categories also enriched categories?

I'm confused about Wikipedia's definition of preadditive categories: In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that ...
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### Proof strategies for dealing with limits and continuity questions.

Background The following Definitions, Theorem, Lemma and Example below are taken from How to Prove it A structured Approach 2nd Edition, by Daniel J. Velleman. The Rule of inference table is taken ...
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### How to define $\Sigma \models \phi$ when $\phi$ is not a sentence? [duplicate]

Let $\Sigma$ be a theory and $\psi$ a sentence. I'm familiar with the notion of $\Sigma \models \psi$, however, lately, I've seen some authors using this notation when $\psi$ is a formula with free ...
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