Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.

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202 views

What does $R^+$ mean?

I'm not sure if it's statistics related but I came over this in my stats related computing assignment. Does $R^+$ (looks like R to the power of plus) mean all positive real numbers? Does it include ...
1
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1answer
101 views

What is the definition of $\sum\limits_{0\leq i\leq m,\text{ }0\leq j\leq n}a_{ij}$

I understand the concept of double summations, at least intuitively, but I'm trying to understand it formally. So, to begin with, I have a question: Is this double summation equality true by ...
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2answers
40 views

Simple notation question

Let A = {2, 3, 4, 6, 7, 9} and define a relation R on A as follows: For all x, y ∈ A, x R y ⇔ 3 | (x − y). Then 2 R 2 because 2 − 2 = 0, and 3 | 0. What does the ...
4
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2answers
115 views

Proving $\bigcup \mathcal{P}(\mathcal{A})=\mathcal{A}$

I'm working on operations on collections of sets and I've run aground. I'm trying to prove that if $\mathcal{A}$ is a collection of sets $\mathcal{A}_i, i=1,2,...$, then ...
2
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1answer
102 views

Difference between colon and membership symbol?

For my discrete math class, my instructor has told us that the following notation is incorrect: $\exists x \in \mathbb{N} | x > 0 \bullet x < 3$ . And we should instead write: $\exists x : ...
2
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1answer
79 views

Style when typesetting functions and operators

I've been making an effort to type all function names and operators in roman font. For example $$\int \operatorname{f}(x) \, \operatorname{d}\!x$$ This was all well and good until I tried to write ...
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1answer
59 views

Is this an accurate way to represent n! using Π?

I recently learned of the $\Pi$ symbol, and was wondering if the following is an accurate way to represent $n!$: $\Pi_{i=0}^{n-1} n - i$
2
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0answers
82 views

Is this symbol $\supset\kern-1.7pt\rightarrow$ commonly used in mathematics?

In Multidimensional Real Analysis I by J.J. Duistermaat and J.A.C. Kolk, the symbol $\supset\kern-1.7pt\rightarrow$ is commonly used. For example, $f: A\supset\kern-1.7pt\rightarrow B$ would mean a ...
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2answers
359 views

Sum Notation with restrictions

I understand normal sigma notation but what does it mean when we place under a sum the restriction that $i + j + k = n$, for example? Is this simply $3$ sums in disguise or is it something else?
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1answer
1k views

Identities for Kronecker delta and alternating unit tensor

How I can prove this equations? Please help me... I can solve it. $$\sum_j\sum_k \varepsilon_{ijk} \varepsilon_{hjk} = 2\delta_{ih}$$ $$\sum_k \varepsilon_{ijk} \varepsilon_{mnk}= \delta_{im} ...
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2answers
57 views

What does $2^H$ mean, where H is finite group?

From Henry Cohn paper: Definition 6.5. Let $H$ be a finite abelian group. An $H$- chart $\mathcal{C} = (Γ, A, B, C)$ consists of a finite set of symbols $Γ$, together with three mappings $A, B, C: ...
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2answers
76 views

An expression which has $10$ different meanings by using some brackets appropriately

My friend taught me the followings without his memory of the answer: He said that it is known that there exists an numerical expression which satisfies the following two conditions : Condition 1 : ...
1
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0answers
256 views

Measure theory, notation

This is from Avner Friedman's "Foundations of Modern Analysis": Let $\mu$ be a measure with domain $A$ and let $E_n$ ($n=1,2...$) be sets of $A$. Then $\mu (\underline{\mathrm{lim}}_{n \rightarrow ...
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2answers
55 views

Are these statements in logic form correct?

Let M represent the set of all Mathematics courses and S represent the set of all students. Predicates: ...
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2answers
77 views

Mathematical notation help.

Is saying !(6 = 4k) a right way to express that 6 is not divisible by 4? Or is there a better more accepted way? I'm writing a proof for discrete math and I need to be sure I'm doing it right.
2
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1answer
138 views

Matrix notation: Does empty space means a bunch of zeros?

I don't understand what is meant with the following notation: I think this means that the first row = 4 2 0 ... 0 second row = 1 4 1 0 ... 0 third row = 0 1 4 1 0 .. 0 etc. Is this correct ?
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1answer
78 views

about powers of gradient operator

Let u be a smooth function and ∇ is the gradient operator in n dimensions such that $\nabla^2 u=\Delta u$ is obvious. However, if we set D=$(\partial_t,∇)$ as a PDE operator in $(n+1)$ dimensions, ...
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0answers
54 views

Is there any way to simplify this expression so that a term only appears once?

I have an expression: $$ \frac{x+z}{y+z} $$ Can I reorganize it so that any variable only appears once?
3
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1answer
48 views

Is there any way to write this number with these limits?

Is there any way to write this number: $$ \frac{17+15\sqrt{5}}{7+15\sqrt{5}} $$ Just using the digits $1$, $4$, $7$, $8$, $9$ once each with the following criteria: A maximum of two square root ...
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0answers
653 views

Definition of multiplicity of a point (in a plane curve)

In the book "Basic Agebraic Geometry I (third edition, 2013)" at page 14 Shafarevich says, about plane curves, what it follows: If $P=(0,0)$ and the leading terms (note:by leading terms I suppose ...
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1answer
195 views

What does $(X_n, Y_n)$ mean? ($X_n, Y_n$ are two sequences of real numbers)

I apologize for this very basic definition question, but I am stumped because my professor uses this notation without any introduction. Verbatim, "Recall that one of the properties of algebraic ...
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1answer
131 views

Why are certain greek characters used predominantly for certain purposes in mathematics?

For example, $\epsilon$ and $\delta$ are used in Real Analysis for proving limits... $\phi$, $\nu$ and $\mu$ were introduced to me in the context of number theory... and then $\pi$ is used for ...
4
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1answer
331 views

Can we define the $L^2$ norm for a vector field $F: \Omega \subseteq \mathbb{R}^d \to \mathbb{R}^d$?

Let $\Omega \subseteq \mathbb{R}^d$ be open and suppose we have a measurable vector field $F : \Omega \to \mathbb{R}^d$ (we consider both the domain $\Omega$ and range $\mathbb{R}^d$ with Lebesgue ...
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1answer
38 views

Notation For $\sup\mathbb{R}$

I understand that $\mathbb{R}$ has no supremum, because the real numbers go on forever. My question is that if I was to write $\sup\mathbb{R}$, would I say it doesn't exist or would I say that ...
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2answers
405 views

When to use congruent vs approximately?

What is the appropriate usage of the congruent ($\cong$) vs. the approximately ($\approx$) symbols?
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3answers
949 views

How to write this in mathematical notation?

I have the following claim: “If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.” I'm supposed to write this in mathematical notation. It's ...
2
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3answers
178 views

What does the notation $\mathbb{P}V$ mean for a vectorspace $V$?

In algebraic geometry, I keep seeing the notation $\mathbb{P}V$ when $V$ is given as a vectorspace. My best guess is that $\mathbb{P}V$ is to mean the projective closure of $V$. But it would be nice ...
3
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2answers
147 views

Why write $1+2+\cdots+n+(n+1)$, not $1+2+\cdots+(n+1)$?

Can someone explain to me this : Why we use both, "n" and "n+1" in the third stage if math induction (where we check if statement holds for "n+1". I'll give an example Prove that ...
0
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1answer
111 views

Notation confusion in the Wikipedia article on the Law of Large Numbers

In my infamous attempt at mastering (at my humble level) the "art" of probability and statistical theory, I was reading the Wikipedia article on the Law of Large Number and got confused by a couple of ...
3
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1answer
104 views

Notation: “belongs to” with an R subscript

I've run into an expression: $x_i \in_R \mathbb{Z}_q$ – and I wonder what this means. An example paper is here, here's example in Wikipedia. Can anybody help me? Thanks in advance.
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1answer
912 views

Why does $\mathbb{Z}$ denote the set of integers?

ok, So the set of Natural Numbers is denoted by $\mathbb N$ (For Natural) The set of Rationals are denoted by $\mathbb Q$ (for quotient) The set of Complex numbers are denoted by $\mathbb C$ (for ...
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2answers
573 views

How to express this statement into logical notation?

Statement: Natural numbers that are exactly divisible by 2 are not prime. I got this: ∀n ∈ N,¬P(n)∧(n%4) where P(n) is the predicate "n is a prime number" and N is the set of natural numbers. And ...
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2answers
59 views

How to read this logical statement in English?

Statement: ∀n ∈ Z, [(P(n) ∧¬(n=2)) ⇒ O(n)] where Z is a set of natural numbers P(n) is the predicate "n is a prime number" O(n) is the predicate "n is an odd number" I got this, but I don't think it ...
1
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1answer
57 views

Ramsey Theory//Question of Notation

I RESOLVED THIS QUESTION FOR MYSELF, THANKS FOR VIEWING, SORRY We have this notation: $$ \mathcal A \to \mathcal B_k^{n} $$ Which means: \begin{align*} &(\forall A \in \mathcal A)(\forall ...
2
votes
4answers
216 views

Using “$\cdot$” as variable placeholder

I've occasionally come across the use of "$\cdot$" as placeholder for a variable, most recently in a paper on radial basis functions, which were defined as $$ s(\cdot) = p(\cdot) + \sum_{i=1}^N ...
0
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1answer
71 views

What notation is used for the canonical representation of $n$?

I've looked on Math.SE and Wikipedia so far and found nothing. I thought I found what I was looking for, but it turned out to be Euler's Totient function. Taking the canonical representation of an ...
6
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1answer
92 views

Wrong use of function notation $f(n)$

I've recently read in a book about computational complexity theory: $$ O(f(n)) = \{g:\mathbb N \to\mathbb R \cup \{0\} : \exists \xi > 0,n_0\in \mathbb N\;\: g(n) \leq \xi \cdot f(n) \;\: \forall n ...
2
votes
1answer
35 views

What is the notation $M_{n}(\mathbb{R})$?

I'm familiar with $M_{m\times n}(\mathbb{R})$ being the set of all $m\times n$ matrices, but I'm not sure I know what this one is.
0
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1answer
197 views

What does the notation $H\biguplus RH$ mean?

I have some problems understanding the notation used in this question. Let $K:= \left\{P\in GL_{2}\mathbb{(R)}: P^{T}P=I_{2}\right\}, H:=\left\{A_{\theta}=\begin{pmatrix} \cos(\theta) & ...
1
vote
1answer
32 views

How to express normalisation where rows sum to 1 for multiple rows?

I have a list of lists. I want to divide every value in each list by the sum of that list. What I did is: a list of lists S, where each list in ...
0
votes
1answer
44 views

What does $[-1,1]\times\{0\}$ mean?

What does the following notation mean? $$[-1,1]\times\{0\}$$ Does that mean all the elements of the form : $(a,0)$ where $a\in [-1,1]$? PS: $[-1,1]$ is an interval on real line.
0
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2answers
55 views

Complex derivative and partial derivatives, question on notation.

It just struck me that the complex derivative of $f : \mathbb{C} \supset D \to \Delta \subset \mathbb{C}$ at $z_0 \in int(D)$ and the partial derivative w.r.t. $x$ have nearly identical definitions ...
4
votes
3answers
995 views

For all unique notation?

Is there notation for "for all unique..."? For instance, suppose you wanted to consider all distinct $x,y$ in some set $S$. Would we type $\forall !x,y\in S$? Can we use "distinct" and "unique" ...
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2answers
60 views

How should I read this $\land$ notation?

I'm studying for SOA Exam C and I was recently introduced to this notation for a minimum $\land$. For example, if there is an limit for how much insurance can be paid out, that limit is $u$ and then ...
0
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2answers
112 views

difference between mod 7 and (mod 7)

So, I am studying modular division right now, and I want to clarify one thing. $a = b \mod m$ and $a = b \pmod m$ Is the top one $b = mk + a$ ($k$ is an integer) and the bottom one is $a = mk + ...
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1answer
394 views

Clarification on matrix notation subscript and superscript notation

If a matrix C exists in integers $\mathcal{Z}_q^{mxl}$ what does this mean?
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3answers
240 views

Using both Leibniz' notation and prime-notation for a derivative

I am presented with the following task: "Assume that the function $f(x)$ has the derivative $f'(x) = \frac{1}{x}$ and that $f$ is one-to-one. If $y = f^{-1}(x)$, show that $\frac{dy}{dx} = 1$. The ...
0
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1answer
72 views

Using additive or multiplicative notation when showing isomorphism (between a subgroup and its left coset)?

I'm trying to understand the proof of that all left cosets of a group $G$ with respect to a subgroup $H$ are equivalent. http://www.proofwiki.org/wiki/Cosets_are_Equivalent And I'm stuck on a tiny ...
3
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1answer
726 views

What is the meaning of the symbol $\pitchfork$?

I have seen this symbol in the formulation of this question. There, it is said: Let $f: X \to Y$ be a smooth map with $f \pitchfork Z$. I was googling, but I haven't found any answers.
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1answer
67 views

Use of “decrease by” in english with a real < 1

My question is about using "decrease by" without any ambiguity. I'm reading a paper where the authors consider a set $X$ defined respectively to a tree and an operation on the tree updating the tree ...