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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Notation for Subspaces

Is there a proper notation for denoting subspaces? For example, if $U$ is a subspace of some vector space $V$. I would usually just write "the subspace $U \subseteq V$" but I'm wondering is there is a ...
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0answers
39 views

Mathematical notation - defining sets

Building on this question, if the cubeful numbers were defined as follows: $$\Bbb Z_{\{3+\}} = \{a \in \Bbb Z \mid \not\exists b \in \Bbb Z \text{ s.t. } a \neq b^3 \}$$ Would it suffice to say ...
2
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2answers
158 views

Notation for unordered product of sets

Frequently, when referring to the edges of an undirected graph $G=(V,E)$, I want to write that $E \subset V \times V$, which isn't correct since the Cartesian product is ordered and the edges are not. ...
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1answer
86 views

Mathematical notation

Is there some generally accepted notation for squarefree, cubefree, etc. numbers? And is there also some notation for squareful, cubeful, etc. numbers?
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2answers
90 views

How would one prove $[f,[\nabla^2,f]]=-2(\nabla f)^2$?

How would one prove this equation: $$[f,[\nabla^2,f]]=-2(\nabla f)^2 $$ And I'm confused that $\nabla f\nabla f$ equals $(\nabla f)^2$ or $\nabla(f\nabla f)$.
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1answer
33 views

Notation to describe amount of relevant elements in a tuple?

Say, we have the set $A=\{♠,♣,♥,♦\}^3$ and would like to define the following map: \begin{align} f: A &\to \{0,1,2,3\} \\ a &\mapsto \text{amount of ♥'s in the tuple } a \end{align} For ...
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3answers
62 views

What does $f|_V$ mean?

I want to proof: If $W$ is vector space, $U, V \subseteq W$ linear subspaces, and $f : W \rightarrow Z$ is homomorphism, $\ker f \subseteq U$ and $W = U \oplus V$ then $f|_V : V \rightarrow Im(f|_V)$ ...
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3answers
80 views

$\sin^2$ notation and uses of the alternative.

So I was taking my calculus class and I was shocked by the following: Apparently its a convention for $\sin^2(\alpha)=(\sin(\alpha))^2$ As opposed to what I thought made more sense which was ...
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1answer
44 views

How to correctly write this ring theoretic thing?

Im unsure how to write this thing below in a formal way : For an integer $n>2$ Let $F_n(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}.$ Also we have $x^n = 1$ and $1 + x + x^2 + ... + ...
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0answers
105 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
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0answers
34 views

Counting matches in a set

I wish to express the number of elements in a set that meet a certain condition $P$, and I feel silly writing: $X = \{x_1,x_2,x_3...\}$ $\sum\limits_{a\in\{y\in X : P(y)\}}{1}$ Though I do think ...
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2answers
87 views

Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$ X\times Y \to X \quad\text{ and }\quad X\times Y \to Y. $$ Usually I denote them by $\pi_X$ and ...
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1answer
67 views

What does this set actually look like? (predicates)

I am learning set theory right now and I am struggling to get to grips with definitions of sets involving predicates. For example, can someone tell me what "typical" elements look like in this set? ...
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4answers
73 views

Question of trig formatting

Is there a difference between the following: $$\sin^2x$$ $$x\sin^2$$ How about: $$\sin(x)$$ $$\sin x$$ I'm new to trig and I've been confused on the formatting involved in trig, whether something ...
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1answer
53 views

Is there a commonly accepted notation for algebraic numbers?

In this question I needed a way to denote an algebraic number using a polynomial equation it satisfies and its isolating polynomial. Because I am not aware of any commonly accepted notation for this, ...
2
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1answer
56 views

Defintion of $\ell^\infty$

I have come across the space of bounded sequences denoted as $\ell^\infty$ in my course, but not a clear, concise definition. I have seen sometimes when these includes sequences in $\mathbb{R}$ that ...
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1answer
311 views

How to convert parentheses notation for trees into an actual tree drawing?

Trees are usually drawn as a set of objects connected by edges. But sometimes one sees a non-graphical, parentheses-based notation, like on the example below. What does the indentation mean in such ...
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2answers
68 views

Definition of $[G:C_G(x)]$

What is the meaning of $[G:C_G(x)]$ in group theory? Is this equivalent to $\frac{|G|}{|Z_G(x)|}$, or to $|Z_G(x)|$?
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1answer
35 views

Notation: $F^{*m}$ for field $F$

Quick question (which is surprisingly hard to Google): if $F$ is a field, what is $F^{*m}$? I suspect it's the $m$-th powers of the elements of the multiplicative group $F^*$, so $\{ x^m : x \in F^* ...
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0answers
20 views

Good notation for many random points approximating an area.

I'm trying to say that as the number of random coordinate points points you plot approaches infinity, it is equivalent to an area integral where each point is an infinitesimally small $\mathrm{d}x ...
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1answer
17 views

What does it mean to randomly choose an integer from a constant?

In this paper on pg. 1241 under section 2.3 "The Elect Protocol" 2nd paragraph the author says Each party samples a random value $x_i$ from [n/k]. What does that mean? If there are n parties ...
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1answer
129 views

The ring hom $\mathbf{Z}\rightarrow A$

Let $A$ be a ring with idenity. Is there a standard notation for the canonical ring hom $\mathbf{Z}\rightarrow A$?
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1answer
35 views

summation index notation, specify all variables?

I am reading lecture slides for a logistics course and for one of the Linear Programming contrainsts, the summation is written as follows: $$ \sum_{i \in I} X_{ij} $$ and $$ \sum_{c \in C} Y_{jc} ...
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0answers
30 views

Skip specifying sets under summation signs

I have a couple of summations that all have the notation below them that s is an element of S, k is an element of K, etc. The set S or K is not given but assumed to be generally known. $$ \sum_{k ...
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1answer
49 views

How to characterize the set of all real functions defined on $X$.

Let $X$ be an arbitrary set. I consider the set of all real functions defined on $X$. I know that this is usually denoted by $\mathbb{R}^X$. However, I am interested in characterizing each point of ...
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2answers
99 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
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2answers
60 views

What does power of '+' in $\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$means?

I encountered the following equation in a paper. $$\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$$ What does the power of '+' mean? The paper can be viewed at: ...
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1answer
55 views

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1$

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1, f[0,1,2]=2$ (The values are exact, which is pretty confusing too, if they are refering to intervals) This question is from a ...
4
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3answers
974 views

'Does not necessarily equal' symbol

What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in ...
2
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1answer
177 views

Why do we write $f : X \rightarrow Y$ as opposed to $f \in X \rightarrow Y$.

I've always been taught to write $f : X \rightarrow Y$ as opposed to $f \in X \rightarrow Y$. This seems weird though, since $X \rightarrow Y$ can be viewed as the set of all functions with source $X$ ...
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4answers
264 views

Mathematical notation around the world

What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when ...
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2answers
140 views

Unable to understand combination of quantifiers and set notation

I know what universal and existential quantifiers are but following is confusing,may be its comibination of set notation and quantifers. What does the following statement means? ...
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1answer
45 views

Is it generally preferred that empty products are gotten rid of where possible?

Is it generally preferred that empty products are gotten rid of where possible? For example: Stewart's structure theorem says that for a positive integer $n$, every positive integer $\leq n$ has a ...
2
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2answers
231 views

About some notation of the derivative

I'm currently Rudin's Principles of mathematical analysis, there is this definition of the "partial derivative". If $f$ maps an open set $E\subseteq R^n$ into $R^m$ and $\{e_1,...,e_n\}$ and ...
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0answers
56 views

Number rings and (round) parentheses versus (square) brackets [duplicate]

Is there a reason for the difference in the use of parentheses versus brackets as used in algebraic extensions. For example, when the field rational numbers ${\mathbb{Q}}$ extended with $i = ...
0
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2answers
30 views

Is this element-of_{ij} - looking symbol the Levi-Civita symbol?

I'm reading this formula: from a page Is the symbol that looks like an element-of symbol with two indices i and j the Levi-Civita symbol? Mathematics is my weak-side so I'm not sure. Actually I ...
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2answers
194 views

Notation of a function that maps a random element

Let there be a functions $f$ and $g$ such that, $$f:A \times B \mapsto \Re$$ $$g: B \mapsto A$$ where $\forall b \in B$, $g(b)$ is some $a$ such that, $\forall a' \in A, f(a,b) \geq f(a',b)$. (This ...
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1answer
47 views

The meaning of $Z_a$, where $Z$ is a partitioning of $A$ and $a \in A$

I am unsure about the common usage of subscripting a set with something, more precisely something which might be a member of that set or some other set (as opposed to, say, subscripting a set with an ...
0
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1answer
987 views

What the symbol $\subseteq$ represents generally? [duplicate]

My book says that $\subset$ is used to represente any subset, proper or improper, needing in this case to show the anti symmetric property of sets. ($A = B \iff A \subset B \, \, \wedge \,\, B \subset ...
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1answer
108 views

Name for a category

Is there any name or notation for this category? Let $U$ be a set. By "function" I will mean a function $U\rightarrow U$. objects are functions; morphisms from a function $A$ to a function $B$ are ...
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1answer
146 views

Time complexity in terms of theta notation [duplicate]

sum= 0; for (i = n; i > o; i = i/3) for (j = 0; j < n^3; j++) sum++; what is the time complexity (in Θ- notation) in terms of n? so far, i've gotten to this point: The ...
4
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0answers
224 views

${\bf R}$ and ${\mathbb R}$ [closed]

My questions are about ${\bf R}$ and ${\mathbb R}$ Some of books use ${\bf R}$ for a set of real numbers. For example Apostol's Mathematical analysis, Adams and Guillemin's measure theory and ...
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1answer
315 views

Time complexity function in terms of theta notation

sum = 0; for (i = 0; i < n; i++) for (j = 1; j < n^3; j = 3*j) sum++; what is the time complexity (in $\Theta$-notation) in terms of ...
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2answers
595 views

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto ...
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2answers
80 views

Notation for “parallel” morphisms in a diagram

Suppose $f\colon A\to B$ and $g\colon A\to B$ are possibly-distinct morphisms. How do I stick them both in a diagram (along with, e.g., their (co)equalizer) without suggesting that they are equal?
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2answers
41 views

notation question re: function space

This is a quick notation question: when one writes $X: C[0,\infty) \to \mathbb{R}$, what does that mean exactly? Is $C[0,\infty)$ the space of continuous functions with a domain of $[0,\infty)$ and ...
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1answer
16 views

Unfamiliar with notation : $S \subseteq [d]$ where {$d, w_{1},w_{2}, …, w_{d}$}

What does $S \subseteq [d]$ mean in the context of {$d, w_{1},w_{2}, ..., w_{d}$}? I don't get what [d] stands for.
3
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0answers
44 views

Complex root notation

Is there a standardized way to distinguish between real and complex roots? In other words, is there a convention about how to I formally write that I expect $\sqrt[3]1$ to be solved in $\mathbb C$, ...
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2answers
254 views

Group theory notation

What does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? And ...
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1answer
13 views

schematic representation of circular permutation in a set?

I would like to represent digits in a set in a way that the set emphasizes the order of the digits schematically in the set in order to avoid confusion with a set of combinations. For example, how ...