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

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1answer
21 views

Radical function in two (or more) variables

I'm reading a paper where the author uses the word radical function for a function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$. I understand the definition of a radical function if $n=1$, but what if ...
0
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1answer
42 views

Useful definition of limit?

Is the following definition of a limit a useful one? Does it make sense? Why/Why not? $lim_{x\rightarrow a} f(x) = b \Leftrightarrow \forall \varepsilon > 0 : 0<|x-a|<\varepsilon ...
8
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3answers
282 views

Why is the universal quantifier $\forall x \in A : P(x)$ defined as $\forall x (x \in A \implies P(x))$ using an implication?

And the same goes for the existential quantifier: $\exists x \in A : P(x) \; \Leftrightarrow \; \exists x (x \in A \wedge P(x))$. Why couldn’t it be: $\exists x \in A : P(x) \; \Leftrightarrow \; ...
7
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6answers
291 views

Is $540^\circ$ a straight angle?

The usual definition of a straight angle is a $180^\circ$ angle. however, because a $540^\circ$ angle is also the same shape, is it a straight angle as well?
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1answer
163 views

Why does an odd number plus one, not necessarily entail it being even?

Why does an odd number plus one, not necessarily entail it being even? For example, $\sqrt{5} + 1$ is not even.
2
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0answers
72 views

Types of definitions

My question is about terminology. Consider the following two types of definitions of a circle: A circle is the collection of all points at the same distance from a given point. A circle is the ...
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1answer
62 views

Limit as universal arrow

I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book. Let us begin with some notation: Let $C$ be a category, and $J$ be an index ...
0
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0answers
93 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...
3
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1answer
51 views

What is the difference between a function and a functional?

When we read Functional Analysis it is said that it is a study of Functionals. I want to know how is it different from the study of functions.
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1answer
19 views

meaning of $f_{\chi_{E}}$

given $(X,\mathcal{M})$ a measurable space, I Have $E \subset X$ and $\chi_{E}$ is an indicator function. then what is meant by $f_{\chi_{E}}$ ? I am not very clear with this notation and meaning.
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1answer
42 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
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0answers
103 views

Why is the conditional probability treated as a definition in Kolmogorov's probability theory?

The conditional probability is defined as: $$P(A|B) = \frac {P(A \cap B)} {P(B)}$$ given that $$P(B) \neq 0$$ This is achieved based on our intuition, along with the Venn diagram description of the ...
2
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0answers
32 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
0
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1answer
88 views

Help with the definition of ordinal sum

"Given two ordinals $\alpha$ and $\beta$, let $A = (\alpha$ x {0}) $\cup$ $(\beta$ x {1}). Then, define a well ordering on A by: $(v, i) <_{A} (\tau, j) \iff (i \lt j) \lor (i = j$ $\land$ $v ...
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3answers
2k views

How to write “let” in symbolic logic

How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is: $$ x := a ...
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0answers
44 views

Definition - limit of a sequence - uses “rank”?

I have the following definition in my book, and was confused as to the context of the word "rank" here. The definition is as follows: A sequence $(u_n)_{n∈N}$ has limit $l ∈ R$ as $n → ∞$ (we also ...
3
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1answer
127 views

What is a endomorphism of vector bundle?

Quick question: When we say $f:E\to E$ is an endomorphism of the vector bundle $\pi:E\to M$, do we require that $f$ maps each fiber $E_p$ to itself, or it could be to another fiber $E_q$? I couldn't ...
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3answers
262 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
2
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2answers
164 views

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
2
votes
3answers
478 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
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0answers
39 views

A quick question on the limsup of sets and logical connectives

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we ...
3
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1answer
66 views

Subtlety in the Definition of Limit Point

In my time studying mathematics I have always found some subtle confusion with the definition of a limit point. I know it's possible for different definitions to yield the same results, and the ...
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0answers
22 views

What's actually $S^k(M)$?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algebra of $M$. Then, what is $S(M)$ (symmetric algebra and $S^k(M)$? Some articles define $S(M)$ as a quotient of ...
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0answers
30 views

Unique Union Problem

Given a the set $a = \{1,...,n\}$ and let $b$ denote a set of subsets of $a$. Find a subset of $c$ of $b$ so that the union of all subsets in $c$ is equal to $a$ and the intersection of any of the ...
0
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2answers
28 views

definition of negative binomial in probability karr

The book defines the probability of the negative binomial as: $$P\{X=k\}={{k-1}\choose{n-1}} p^k (1-p)^{k-n}$$ but where does the ${k-1}\choose{n-1}$ come from? It's quite different to wikipedia's ...
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0answers
54 views

Definition of some terms of transformation geometry

Recently I was studying transformation geometry from problem solving strategies . I liked the subject but could not understand some terms. Please anyone help me--- What is isometry? What is ...
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0answers
62 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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1answer
91 views

How to call “equivalent-looking” vertices in graph..?

In the above figure, the vertices expressed as blue dots are "equivalent-looking." Although my expression is somewhat ambiguous, I believe one can simply answer it. How can we call such vertices? ...
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1answer
29 views

Intuition behind a particular definition regarding cycles of partitions

I'm reading this paper on cycles of partitions, and was wondering if anyone could motivate the last condition in the definition of the sets $M_n$ in terms of the partitions being examined. In ...
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2answers
1k views

Definition of a Minimal Set

A few times while studying math I have encountered the notion of a "minimal set". For example, given some set of subsets, what is the "minimal" sigma algebra generated by that set of subsets? Or, in ...
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0answers
46 views

What is the “minimal” structure in which points and lines are defined?

Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective ...
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2answers
69 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
0
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1answer
71 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
0
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1answer
24 views

interpreting the curve of intersection

I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of ...
-2
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4answers
96 views

Find the derivative of $f(x) = \sec(x)$ without the quotient rule [closed]

Part of an analysis assignment I have: " Given $f(x) = \sec(x)$, compute the derivative of $f(x)$ by using the definition of derivative. (Note that $\sec(x) = 1/\cos(x)$ and $(\cos(x))' = ...
2
votes
2answers
86 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
20
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7answers
2k views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
1
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1answer
28 views

Definition for trigonometric function with a different “system”

For fun, I decided to create a sort of "intuitive" (for me, anyhow) approach to degrees and such. As I can recall, degrees are based on (the Mesopotamians?)'s base $60$ math. I've read that radians ...
3
votes
8answers
156 views

How can expressions like $x^2+y^2 = 4$ be defined?

I'm wondering how to define the expression(?) $x^2+y^2 = 4$, because I realised it's not a function because it cannot be expressed in terms of $x$ or $y$ alone. Is it even called an expression? Of ...
2
votes
1answer
76 views

Cotangent space explicit definition

Given a tangent space $T_xM$, where $M$ is a differentiable manifold homeomorphic to $\mathbb{R}^n$, we have the cotangent space $T^{*}_xM$ defined as being the set of linear functionals $\eta: T_xM ...
3
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1answer
35 views

How do I define a block of a $(0,1)$-matrix as one that has no proper sub-blocks?

I'm struggling to come up with a definition of a "block" in a $(0,1)$-matrix $M$ such that we can decompose $M$ into blocks, but the blocks themselves don't further decompose. This is what I've got ...
0
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3answers
55 views

Does Stars and Bars or the binomial coefficient represent binary sequences?

Does Stars and Bars or the binomial coefficient represent binary sequences? With the binomial coefficient we can calculate all the paths on a grid with moving up or right, that's like defining up to ...
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1answer
56 views

What is a “right” automorphism?

Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads: If an automorphism of ...
0
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0answers
62 views

Hausdorff Distance on Fuzzy Sets

I'm trying to define the Hausdorff distance between two fuzzy sets in terms of non-fuzzy sets. Is this a viable definition? How can I show that this reduces to the Hausdorff Distance for non-fuzzy ...
0
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1answer
104 views

n-tuple function definition

I've read this definition for an hour now and I cannot piece it together abstractly. To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the ...
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1answer
113 views

How many teams of $5$ players out of $15$ girls and $10$ boys can be formed with at least $2$ boys and $2$ girls [with complement]

How many teams of $5$ players out of 15 girls and 10 boys can be formed with at least 2 boys and 2 girls? The solution has to be with complement. This is related to: How many ways to assemble a ...
1
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1answer
20 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
0
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1answer
77 views

What is the difference between an antiderivative and an integral?

In my textbook, it states the fundamental theorem of calculus as follows: If $f(z) $ has an antiderivative $F(z)$, then $\int^{z_2}_{z_1} f(z)dz=F(z_2)-F(z_1)$. There isn't a definition of what an ...
5
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2answers
114 views

Degree of a map $S^0 \to S^0$

By definition the degree of a map $f: S^n \to S^n$ is $\alpha \in \mathbb Z$ such that $f_\ast(z) = \alpha z$ for $f_{\ast}:H_n(S^n) \to H_n(S^n)$. What is the definition of the degree of $f: S^0 \to ...
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1answer
94 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...