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Questions tagged [definition]

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

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Confusion over the definition of “model”

In my question yesterday I asked about the definition and usage of the word "model", for which I was told the following definition: A formula of propositional logic is true under an interpretation ...
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Definition of Clifford Algebra

Cliffard algebra defined by relation: $x*y+y*x=g(x,y)1$, where g(x,y) is bilinear symmetric form. What does mean $g(x,y)1$, why it's not just $g(x,y)$, without the identity?
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Collection of Nonempty Subsets of $\mathbb{Z^{+}}$ a Class or Set? [duplicate]

In ZFC, is the collection $\mathcal{N}$ of nonempty subsets of $\mathbb{Z^{+}}$ a class or set? Please correct me if I am wrong, but so far as I understand the notion of class, a class is a collection ...
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Lebesgue Integration in practice on a bounded continuous function.

Suppose a continuos lebesgue measuable function $f$ that is (non negative) bounded above by $M$. Define a sequence (similiar to standard represtation in counting measure) $\displaystyle f_m=\sum_{i=0}...
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Interpretations and models in propositional logic

I am trying to understand model vs. interpretation of sentences vs. theories, with respect to a propositional calculus. Is a model of a wff just a sentence that is true under some interpretation? ...
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What is the order of an unramified character?

Let $K$ be a local field and $G_K$ be its absolute Galois group. A character is a group homomorphism $\phi: G_K \to \mathbb{C}^*$ with finite image, i.e. $|\phi(G_K)| < \infty$. It is unramified if ...
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Principal Bundle definition

Let $G$ act on smooth manifold $M$. If the action is free $Stab(G) = \{ e\}$, then $O(p) = Orbit(p) \approx G$. In the definition of principal bundle with $(E,\pi, B, G)$, then let $G$ acting on $E$ ...
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Understanding the definition of the limit of a sequence

Consider the "formal definition" here https://en.wikipedia.org/wiki/Limit_of_a_sequence. I checked some references and this is often precisely the definition in all words and terms used in this ...
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Why do we need the following set to have an infinite set?

By Dedekind's definition: Definition: We say that some set $S$ is infinite if there exists an injection $f:S\rightarrow S$ such that $\operatorname{Im}(f)\neq S$ (or $f(S)\neq S)$. Now, the book ...
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What is meant by “the group $\langle a \rangle$ acts on the set $A$ by conjugation?”

A proof for the fact that an abelian group of order 15 must be cyclic states is given here: https://math.stackexchange.com/a/208801/115703 In this proof, what is meant by "The group $\langle a\...
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Injective and Projective Modules

There are several equivalent definitions of projective and injective modules over a ring $R$ (with unity). However, I didn't find anywhere the justification for using words injective and projective ...
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Binomial's extensions

In some questions about polynomials, I've found the following formula: $\binom{a}{k}=\sum_{j=0}^k\binom{a-b}{j}\binom{b}{k-j}$ This can be gotten of the coeficient of $x^k$ in $(1+x)^a$ and in $(1+x)...
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Understanding completion of metric (vector) spaces

I am wondering if I have understood the consept of completion of a metric/normed space correctly. As I have understood the completion theorem, it is: $$\textbf{Completion theorem for metric spaces}$$...
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Definition of span

On an old midterm exam, my professor requested the students prove that The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear ...
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Terminology: Path, Curve, Graph

Given a "space" X, at least topological so that continuity has meaning (perhaps a vector space, $R^n$, or the complex plane), and a closed interval I of the real line would it be reasonable to assert ...
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Isn't the term vector space a misnomer?

As a Physicist, I know: A vector is defined to be an object with a single index which follows certain properties related to rotation in the space in which it resides. But when we study abstract ...
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How to interpret invariant subspaces in the definition of irreducible representation?

I have this sentence in my notes: A representation $D$ from a group $G$ to matrices $M(V)$ acting on a vector space $V$ is irreducible if and only if $V$ has no non-trivial invariant proper ...
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What is a continuous random variable? A Collection of definitions

Although this is a question about what's a continuous random variable, it seems that there are at least 2 definitions being used. The Distribution function is continuous. There exists a non-negative ...
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Algebraic definition(s) of constructible numbers [closed]

Another question which might be considered opinion-based, sorry for that. While the extensions $$\operatorname{Quot}(\mathbb{Z}) = \{\frac{p}{q}\ |\ p,q \in \mathbb{Z}, q \neq 0\} =: \mathbb{Q} $$ ...
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How to define a linear transformation

Let T and U be the linear operators on $R^2$ defined by: $T(x,y)$=$(y,z)$ and $U(x,y)$=$(y, 0)$ Give rules like the ones defining T and U for each of the transformations (U+T),UT,TU,$T^2,U^...
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derivative question confusion

if we have a function f where $f(x)=x^2$ then the derivative function $f'$ can be calculated $f'(x)= 2x$, and this gives the derivative of f wrt x at some point x in the domain of f. my confusion ...
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Clarification on Lie Algebras Notes: “Let $\mathfrak{gl}(V) = End(V)$”

I wanted to ask about something confusing in my Lie Algebras notes. At one point we define the Lie Algebra $\mathfrak{gl}(V)$ to be the vector space $GL_n(V)$ with the bracket $[x , y ] = xy - yx$. (...
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Categorical characterizations of ring properties

Looking at the interesting list of ring properties that are inherited from a ring $\mathcal{R}$ by its polynomial ring $\mathcal{R}$[X] and remembering a question I once asked I want to repeat the ...
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Definition of Hausdorff space and please check my proof

I'm not sure about Hausdorff space property. It's all point $x,y\in X$ with $x\neq y$ there exist open sets U containing $x$ and V or just some point $x,y$ Here this is my problem that I want to ...
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Where is there a list of formal, accepted definitions of all mathematical terms?

I read on the wikipedia page for real numbers that the simple definition given is not up to the standards of formal, rigorous mathematics. Where, then, do we have a list of definitions for every ...
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Word and symbol for putting two sets in to a larger set?

After doing some work with sigma-algebra, I'm stunned that I can't recall ever seeing any name or notation for this. However, consulting Google and Wikipedia has turned up nothing. What I'm looking ...
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1answer
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What does $x \sim X$ mean in probability?

I just want to confirm what $x \sim X$ mean in probability. What does the small $x$ and big $X$ represent? And can we replace $\sim$ with $=$. Disclosure: I had read many probability textbooks from ...
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The logic behind adding this sentence to the second condition in this definition of isomorphism

Below is the definition of isomorphism quoted from my textbook. First, we introduce relevant definitions: An $n$-ary relation $R$ in $A$ is a subset of $A^n$. Then we write $R(a_1,\cdots,a_n)$ to ...
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Closure of a constant

Consider a type $T$, and a set $S$ containing elements of type $T$. An object $f$ of type $T→T→T$ (or $T^2 → T$) is a function and it is closed under $S$ if any two elements of $S$ applied to $f$ ...
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Can we define complex path integration in a particular way.

Take a curve $z(t) = x(t) + i y(t), a \leq t \leq b$ in the complex plane. Consider a partition $Q= (a = t_0, t_1, \dots, t_n = b)$ of $[a,b]$. Associate a partition $P = (z(t_0), \dots, z(t_n))$ ...
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What does “definite” mean for a bilinear form over a finite field?

What does it mean for a symmetric bilinear form over a finite field to be definite? Most sources (e.g. Wikipedia) only define definiteness for a form over $\mathbb{R}$ or $\mathbb{C}$. Yet I have ...
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Definition of trace in Bourbaki

Bourbaki, General Topology, p. 61 (1966) What is the definition of trace in the following Proposition? Proposition 8. Let $\mathcal{F}$ be a filter on a set $X$ and $A$ a subset of $X$. Then the ...
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Why do the concepts of “zero divisor” and “unit” have so different names?

I wonder why the concepts of zero divisor and unit have so different and unrelated names, even though their definitions are in perfect analogy: $x$ is a zero divisor when there is a $y$ with $x\...
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Being-part-of: A neglected algebraic concept?

Algebra, esp. ring theory is about the relationship between addition and multiplication. One concept of exceptional importance is the relation of being a $k$th root of a number (with $k\in \mathbb{N}$ ...
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What is the reason for restricting codomain in such a way?

So I see that usually in real analysis limits are defined only for functions with codomain R, and even for more general definitions of limit, e.g. for metric spaces, I usually see that limit is ...
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Fréchet derivative; open set

I have some problems in understanding the definition of the Fréchet derivative of an operator $F: X \rightarrow X$. In fact, most authors report that $F$ must be defined in some open neighborhood of ...
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Intuitive ways of thinking about reducible nonnegative matrices?

The definition I am thinking of is A nonnegative matrix $A$ is said to be reducible if there exists a aprtition of the index set $\{1,2,\dots,n\}$ into nonempty disjoint sets $I_1,I_2$ such that $...
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What is a primitive character of a Galois groups of a finite cyclic extension of local fields?

Let $K$ be a local field and $F/K$ be a cyclic extension of degree $n$, meaning that $n = [F:K]$ and $\operatorname{Gal}(F/K) \simeq C_n$ is cyclic. In the proof of Lemma 2 of the paper "Euler ...
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1answer
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Expected value and dimensional analysis

The standard definition of expected value $E(x)$ of a continuous distribution $\rho(x)$ is given by $$ E(x) = \int_\mathbb{R} x \rho(x) dx $$ Expected value is exactly what you might think it means ...
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4answers
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Euclidean Algorithm Proof

My lecturer gave us the following side note when explaining the euclidean algorithm in class. Eucledian Algorithm: Let $a$ and $b$ be natural numbers, then there are integers $m$ and $n$ such that ...
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Definition of rooted graphs

Wiki defined a rooted graph as a graph in which one vertex has been distinguished as a root. What do they exactly mean by a root? Is it when every other vertex is the extremity of a path coming from ...
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Various definitions of Order Embedding and Order Isomorphism

From my textbook, Wikipedia, and other sources from the Internet, I have come across various definitions of Order Embedding and those of Order Isomorphism, so I'm very confused about them. My ...
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Two definitions of order embedding, and the reason why one is correct and the other is not

I have two definitions of order embedding and I present the reasoning for their difference below. Please check if my understanding is correct! Thank you so much! Definition 1: Let $(A, \prec_1)$, $...
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Formal definition for when one thing simulates another?

I've been trying to figure out what it means for one thing to "simulate" another. For example, a universal Turing machine can "simulate" any other Turing machine. We might also say that two different ...
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For complex number z = 3 - 2i, why is Im(z) -2 and not -2i?

The general part of the mathematical area of complex numbers is having me stumped, and I feel like as soon as I understand this, it may broaden my understanding of every consequent part of this topic (...
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1answer
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What is $\beta(p_i)$ where $p_i$ is a propositional variable?

Let $\beta \in \mathscr{A}$ and let $P$ be a an infinite set of propositional variables. Is $\beta(p_i)$, for $p_i \in P$ just the value of $p_i$ under the truth assignment $\beta$? Sorry if this ...
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What is a vocabulary in logic

I am reading notes on finite model theory and I came across a notion of a vocabulary. What is it? I tried searching on the internet but did not find anything relevant.
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Matrix property more specific than being triangular — each next row starts with more zeros.

I am looking for a name of a following matrix property: each next row has more starting zeros than the last one. The last few rows can all be full of zeros. I was trying to search for that but found ...
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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 ...
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Confusion about the definition of analytic and singularity.

In my textbook the definition of analyticity is given as A function is said to be analytic in a domain D if f(z) is defined and differentiable at all points of D. The function f(z) is said to be ...