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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27
votes
6answers
2k views

Why does the symbol for the multiplication operation change shape?

Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen ...
2
votes
3answers
34 views

Is there a difference between $y(x)$ and $f(x)$

Oftentimes functions described by $f(x) = 2x+4$, and when this is mapped to the Cartesian plane, $f(x) = y$. This surely implies that $y = 2x+4$. Is there a difference between this and $y(x) = 2x+4$?
0
votes
1answer
31 views

Vector in vector notation

I'm a bit confused as far as notationally differentiating between row and column vectors goes. Suppose I define a column vector $$\boldsymbol{a} = (a_{1}, a_{2})^{T}$$ and another column vector $$\...
3
votes
0answers
37 views

What does $x$ represent?

$x$ can mean: A defined value in $x = 5$ An unknown, TBD. value, in $3x = 2+4$ A variable in $y(x) = x+4$ A sum of numbers in $\int_0^2 \mathrm{d}x$ Are there any other examples?
2
votes
0answers
60 views

Is this notation common in Calculus?

Okay this is going to be quite a stupid question, but to me this seems... wrong, or at the very least not completely correct. In the material I'm reading there's a part that states that $y$ evolves ...
0
votes
2answers
32 views

Set notation for unordered cartesian product

In the question unordered cartesian product an shorthand notation for the unordered cartesian product was discussed but without any standard notation. So my question is what would be the explicit ...
0
votes
2answers
22 views

Summation Notation Question in McMillan's Theorem Proof

Let me preface by saying that this question does not pertain as much to coding theory, as it does to mathematical notation. Every symbol in this question is a natural number. Anyhow, I am currently ...
0
votes
1answer
26 views

Notation for set of unit vectors

Is there a standard notation for the set of unit vectors $\{\vec v\ :\ |\vec v|=1\}$?
17
votes
10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
0
votes
3answers
24 views

How to read this ArgMax definition in plain english

I was reading on Wikipedia about Argmax (https://en.wikipedia.org/wiki/Arg_max) and they gave the following equation. While I get most of this line, how would you read the following in plain English? ...
1
vote
3answers
41 views

How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way. What is a popular way? In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?
11
votes
3answers
456 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\...
-1
votes
1answer
31 views

If G and H are two gaphs then what does $G \Delta H$ indicate in graph theory?

I came across this notation in a book titled " Combinatorial Optimization Theory and Algorithms" by Bernhard Korte and Jens Vygen.
42
votes
12answers
4k views

Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
6
votes
1answer
249 views

A question about co-exponentials

An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$ or equivalently, as the terminal object of $...
1
vote
2answers
38 views

Notation for a sum over a set of variables

I have a vector of variables $y=(y_1, \ldots, y_n)$ whose elements are either zero or one. I would like to express the sum over all variables belonging to a subset $S$. For example, if $n=4$ and $S=\{...
3
votes
1answer
10 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
3
votes
3answers
51 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
0
votes
0answers
31 views

What does the '#' sign mean in elliptic curves?

My question is regarding specifically elliptic curves. I have seen the notation $\#E(\mathbb{F}_{q})$ used over and over again (especially in the description of Hasse's theorem). I know that sometimes ...
0
votes
1answer
32 views

Proper notation for motion integration

Say you have a projectile where at $t=0$, $ v = 0 $ and $ x = 0 $. Given $ \ddot x = -4$, in order to find $ \dot x $, we must integrate $ \frac{dv}{dt} $ as follows: $$ \frac{dv}{dt} = -4 \...
1
vote
0answers
21 views

derivative and curl vector notational difference

This must be a really trivial question, and I am not sure I am allowed to post these here. Just wanted to know but didn't know what to search. What is the difference between $\overrightarrow {dr}$ and ...
2
votes
3answers
62 views

How $f:[a,b]\rightarrow[c,d]$ should be read?

I found it in a book but I don't know what the ":" means. What does this expression mean?
3
votes
2answers
66 views

What does $\mathbb{Z}/n \mathbb{Z}$ mean in abstract algebra?

If you look at this wiki page under the image on top of right hand side, you see $\mathbb{Z}/ \mathbb{8Z}$. What does it mean and give example if possible please thanks.
0
votes
0answers
31 views

Distinguish between constant function and a constant. [on hold]

The constant function $\begin{array}&y: &\mathbb{R}\to \{c\}\\&x \mapsto c\end{array}$ and the constant $y = c$ are often simply written as $y = c$ and it's not always ...
-1
votes
2answers
79 views

Why is $\wedge$ a minimum and $\vee$ a maximum? [closed]

Why does $\wedge$ denote a minimum and $\vee$ a maximum? Where did this notation come from? I keep getting them mixed up because to me, $\wedge$ should be a maximum: it's a hill, or a curve reaching ...
0
votes
2answers
25 views

Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
2
votes
1answer
23 views

Inverse relations notation (not a function) [closed]

Even if $f:X\to Y$ is not a bijective function, can I still notate the inverse relation of $f$ as $g:Y\to X$?
0
votes
0answers
18 views

Is there an accepted notation for the monoid of linear polynomials?

Is there an accepted notation for the monoid of linear polynomials (with addition as the operation) with coefficients from some ring R? Like $2p+3$, where $p$ and the identity generate the monoid ...
7
votes
4answers
141 views

Confused about notation “:=” versus plain old “=” [duplicate]

Relating to sets, I find the following in a text book: "...the set S := {1, 2, 3}". The book has an extensive notation appendix, but the":=" notation is not included. What exactly does ":=" mean, and ...
0
votes
1answer
15 views

Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$?

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though ...
0
votes
0answers
31 views

How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
4
votes
4answers
64 views

I want to write “$x,y > 0$”.

I want to write "$x,y > 0$". Can I do this? Or do I have to write "$x > 0$ and $y > 0$"? Which one is the proper way to write in maths?
0
votes
1answer
15 views

Is it possible to unify these two expressions into one?

I have the following expressions $\forall n \in \Bbb N$: $E = f(n)-1$ if $n \gt 1$ $E = f(n)+1$ if $n = 0$ I would like to have only one expression like this: $E = f(n)+$(some nice notation able ...
-1
votes
1answer
61 views

Why is the notation for irrational number not mainstream? [closed]

In everywhere you see the symbol for the set of rational number as $\mathbb{Q}$ However, to find actual symbol to denote the set of irrational number is difficult. Most people usually denote it as $\...
2
votes
3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
1
vote
1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...
1
vote
1answer
25 views

Notation for minimum

I'm reading Huber's Robust Statistics right now, and at the beginning of Chapter 3, he writes the following notation: $\sum \rho(x_i;T_n) = \min!$ Similarly, a few lines down, he writes: $\sum \rho(...
18
votes
6answers
4k views

How to represent “not an empty set”?

I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians? My idea is: $|A| > 0$ However, using the emptyset $\emptyset$ ...
2
votes
1answer
34 views

Understanding the notation of a paper

I am reading a paper on Algebraic Number Theory that says If $p$ divides the discriminant of polynomial $f$ $r$ times and there is the factorization into irreducibles $$f(x)\equiv g_1(x)\dots g_r(...
1
vote
0answers
38 views

How do I write a function that maps a variable to a set?

I have a function $\Gamma$ that maps elements from $N$ to a (possibly empty) subset of $N$. The number of elements in the resulting subset depends on which element of $N$ we are dealing with, i.e. $\...
2
votes
0answers
35 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
0
votes
0answers
13 views

Meaning of notation $L_{subscript}$ in ridge detection.

In the wikipedia article on ridge detection, it says "let $L_{pp}$ and $L_{qq}$ denote the eigenvalues of the Hessian matrix \begin{pmatrix} L_{xx} & L_{xy} \\ L_{xy} & L_{yy} \end{pmatrix}...
2
votes
1answer
57 views

$T*T$ Notation and proof

Let $T:H\to H$ be compact where $H$ is a Hilbert space and let $T^*$ be the adjoint operator of $T$. Prove that $T^*T$ is compact and self adjoint and that the eigenvalues of $T^*T$ are nonnegative. ...
0
votes
0answers
16 views

Visualization of residual sum of squares in matrix notation

I am trying to understand how to pass from \begin{equation} RSS(\beta) = \sum_{i=1}^n (y_i - x_i^T\beta)^2 \end{equation} to \begin{equation} RSS(\beta) = (y - X \beta)^T (y - X \beta) \end{...
4
votes
2answers
63 views

Notation for “the highest power of $p$ that divides $n$”

If $p$ is a prime and $n$ an integer, is there a standard or commonly used notation for "the highest power of $p$ that divides $n$"? It's a concept that is often used repeatedly in number-theoretic ...
0
votes
1answer
27 views

Reading set notation

I am given a question and am having a hard time understanding how to read part of a question, it reads let $ C^{1}(0,1):= \{f:(0,1) \rightarrow \mathbb{R} \mid f\text{ is differentiable and $f'$ is ...
0
votes
2answers
57 views

What does $\partial_p$ mean?

In the Wikipedia article on ridge detection, there is this symbol $\partial_p$. I know that $\frac{\partial f}{\partial p}$ is sometimes denoted as $\partial_p f$, but I've never seen $\partial_p$ ...
1
vote
0answers
40 views

pound symbol in inequalities

a friend of mine has to solve some equations, but in some of them appears something like a pound symbol. Do you know what it means and how to solve it? or the teacher really lose his wit? Thank You!...
0
votes
1answer
27 views

Comparison of Cartesian and Scalar Notation in Mechanics

In his book on Engineering Mechanics - Statics, R C Hibbeler provides many force problem solutions in both scalar and Cartesian notation (e.g Example 2.5 Chapter 2). It feels like he is trying to ...
0
votes
2answers
33 views

Notation for apply operation to digits of a number

what is the standard notation to represent that a operation has been applied to each digits of a number for example ...