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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0answers
109 views

Implicit arguments in informal math, how to explain?

Let we have three categories $Z$, $C$, and $D$. And let $Z$ have partially ordered Hom-sets each Hom-set having a least element. Let also every object of $D$ be an ordered set and has least element. I ...
1
vote
1answer
20 views

Notation for enumerating a set

Is there a common notation for enumerating a set? For example if $A=\{2,4,6,\ldots,n \}$ is the set of even numbers, I would like to know the notation that enumerates ordered pairs $(e,i) \in \...
0
votes
1answer
13 views

summation notation of elements in several different (sub)sets

If there are two different sets $A$ and $B$, and $A\cap B= \emptyset$, then sum of all elements in both sets might be written as, $$\sum_{a\in A}a+\sum_{b\in B}b$$ What I want to ask is, can I ...
3
votes
2answers
36 views

Notation for conditional set complement?

As far as I know, given $U=\{1,2,3,4,5,6\},A=\{1,2,3\}$ the notation for its set complement is $A^C = \{4,5,6\}$ Is there any sort of notation for a conditional set complement? For example, lets say ...
5
votes
6answers
349 views

Do different notations imply different properties of a number?

I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit. So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the ...
0
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1answer
24 views

Vector magnitude notation

Is the follow equality true? $$\left| \overrightarrow {u}\right| =u$$ I wonder, because on AP Physics formula sheets, sometimes the magnitude of a vector is clearly denoted, while other times the ...
4
votes
3answers
100 views

What is the meaning of $e^{int}$ notation

I have a question and I have a definition $\{\varphi_n(t)=\frac{1}{\sqrt{2\pi}}e^{int}\}$.I know that $n\in \mathbb Z$ for this question. does int means integer? I couldn't find any reference with a ...
2
votes
4answers
75 views

What does the notation $z\in\mathbb{C}\backslash\mathbb{R}$ mean?

I know that $z\in\mathbb{C}/ \mathbb{R}$ means that the domain is the complex plane with the real line removed. What does the notation $z\in\mathbb{C}\backslash \mathbb{R}$ mean? EDIT: Turns out ...
0
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0answers
52 views

Why if $f$ is a function and $x$ an element, what we write $f(x)$ is written $xf$ sometimes.

In the book here, if $G$ and $H$ are two group, the define the homomorphism \begin{align*} \varphi:G&\longrightarrow H\\ g&\longmapsto g\varphi. \end{align*} In fact, if you see a little bit ...
3
votes
4answers
71 views

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} ...
0
votes
1answer
43 views

Is $f\sim g$ an appropriate notation for L' Hôspital Rule?

Let's say $\displaystyle \lim_{x\to0} f(x)=0$, $\displaystyle \lim_{x\to0} g(x)=0$ and $\displaystyle \lim_{x\to0} \frac{f'(x)}{g'(x)}=1$. Conventionally we write $$\displaystyle \lim_{x\to0} \frac{...
0
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0answers
28 views

Theorem $2.15$ of Quiver Representations by Ralf Schiffler.

I'm reading the proof of theorem $2.15$ of Quiver Representations by Karl Schiffler. The author states the following: Let $Q$ be a finite acyclic quiver, $M=(\left\{M_i\right\}_{i\in Q_0}, \left\{\...
0
votes
2answers
24 views

Help with new arrow notation

I am working through a description of gradient descent and I'm having trouble finding the definition of a couple notations, an arrow and a single quote, v→v′=v−η∇C. I normally express a derivative ...
0
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0answers
34 views

Is there any relationship between $\mathcal{l}^p$ spaces and $\mathcal{L}^P$ space?

Due to the similarity of the names, I guessed that there may be some relationship between the two spaces. Is there such a relationship, or is there nothing more to it other than the fact that they ...
0
votes
1answer
60 views

Computing Complex Line Integrals

I'm having trouble understanding exactly how to compute a complex line integral in $\mathbb{C}$. With my understanding of multivariable calculus, I view the line integral of a vector field $F: \mathbb{...
0
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2answers
87 views

What is the difference between $(f^{-1})^{-1}(A)$ and $f(f^{-1}(f(A)))$?

I asked a question Under what condition does $f(f^{-1}(f(A))) = f(A)$? and it totally backfired because people were confused whether $f^{-1}$ is the preimage or the inverse function Let $f: X \to Y$ ...
1
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0answers
33 views

Notation of a function that Maps two sets into a Matrix

Given two sets $P, V$ a function $f(t)$ takes any element that belongs to $ P $ or $ V $ e.g. $ t \in P \cup V$ returns a matrix of $ 2 $ columns and $K$ rows. What is the proper notation to express ...
1
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1answer
64 views

Elements in commutative diagram

The same way I define a function, by explicitly including the image of an element: $$ \begin{aligned} \mathbb{R} & & \overset{\exp}{\longrightarrow} & & \mathbb{R} \\ x & & \...
2
votes
2answers
49 views

$f$ in $f(x)$ as a vector

I might split this question into two, so the first paragraph will contain the main question. Given a linear function $f(x,y)$, is it possible to consider $f$ as a vector? Given the relationship of ...
5
votes
0answers
61 views

Difference between $d\mu(x)$ and $\mu(dx)$

In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$? For example I read $\mu(dx) = \frac{1}{\...
0
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1answer
32 views

How to read the value of -2.0138e-01

Is -2.0138e-01 equal to -0.20138 (or) -0.00138 (or) 0.20138 (or) 0.00138. Not sure how to read the -ve numbers and positive numbers. This website(http://www.easysurf.cc/scintd.htm) shows 0.00138 ...
-1
votes
1answer
31 views

Finding the Time Complexity in Big theta notation [closed]

sum = 0 ; for ( i = 0 ; i < n ; i++ ) for ( j = 1 ; j < n^4 ; j = 4*j ) sum++; How would I go about finding the time complexity in ...
7
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2answers
1k views

What does sgn mean?

I am reading "space filling curve" by Hans Sagan. On page 17, on equation (2.3.11) in the equation, a function sgn is used. What is sgn? To put it into context, in the book it says $h_n=$sgn$(n)[(n-...
3
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1answer
58 views

Notation for “there exists at least 2”?

Let $S = \langle 1,3,5,7,9 \rangle$ be a sequence of a numbers. Then let $Q = \{ 1,2,4 \}$ and $R = {1,2,3}$ be two sets of numbers. Then I have a function $exists(Q,S)$ that should return true if ...
3
votes
1answer
42 views

What does $K(A)$ mean in field theory?

So in my notes it says that if $K\subset L$ is a field extension and $A \subset L$ is a subset then $K(A)$ is a subfield of $L$ containing both $K$ and $A$. It is in fact the smallest such subfield. I ...
1
vote
1answer
13 views

linear transformation notations

I know that L$(V_1, V_2)$ denotes a linear transformation from $V_1$ to $V_2$. What does $L(V)$ denote. My guess would be that it denotes the homomorphism from $V$ to $V$ but I'm not sure.
2
votes
0answers
51 views

Infinite amount of subsequences

Suppose I have a sequence that seems random, increases and diverges to infinity: $A_{n}=\left\{2,5,9,12,15,17,23,25,29,33,34...A_{\infty}\right\}$ Now I want to remove all of the positions in this ...
0
votes
2answers
54 views

How to Read Notation for General Intersection and Union

I am trying to read the following two formulas in English but I am not certain how to, although I do understand their meaning. Union: $\bigcup_{i=1}^{n} A_{i} = \{x\, |\, \exists_i \in I(x \in A_i) \...
1
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1answer
34 views

Notation matter concerning 'or'-elimination

I have to show that $\{(\phi\lor\psi),(\lnot\phi)\}\vdash\psi$ using the following natural deduction rule: I don't know which of these is correct in term of notation: Could you please tell me? ...
1
vote
1answer
57 views

What does “pcr” stand for?

For example, as shown in this OEIS page: $P(n,4)=\frac{2n^3+6n^2-9n-13+(9n+9)\text{pcr}\{1,-1\}(2,n)-32\text{pcr}\{1,-1,0\}(3,n)-36\text{pcr}\{1,0,-1,0\}(4,n)}{288}$
0
votes
0answers
17 views

Notation: density

I know that $B \subset \bar{A}$ means that $A$ is dense in $B$. Is there a symbol that can be used between $A$ (not $\bar{A}$) and $B$ to state the same thing?
3
votes
1answer
35 views

Need hint on induction proof for summation

I have a homework problem to prove the following via induction: $$\sum_{i=1}^n i^22^{n-i} = 2^{n+3}-2^{n+1}-n^2-4n -6$$ The base case is true. I generated the below using $s_k+a_{k+1}=s_{k+1}$: $$ 2^{...
1
vote
2answers
27 views

Converting from standard to functional, Polish and Reverse Polish notation

I wanted to convert the following expression to Functional, Polish and Reverse Polish notation. $$Y =A + \frac{B+ \dfrac{BA}{B+CA}}{A - \dfrac{BC}{B-C+A}}$$ I know how to do Standard -> Functional ->...
1
vote
2answers
90 views

Are integers relevant for every Group?

The definition of the order of an element in a group is: The order of an element $x$ of a group $G$ is the smallest positive integer $n$ such that $x^{n}=e$. Doesn't this definition assume that ...
4
votes
1answer
56 views

What is $\circeq$ used for?

I was looking for a way to put 'des' over the equals sign in latex and stumbled upon this symbol $\circeq$ What are mathematicians using it for?
0
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3answers
74 views

Meaning of mathematical symbol $\pm$

What is the meaning of the $\pm$ symbol in relation to this expression? For example, the perceived area of a circle probably grows somewhat more slowly than actual (physical, measured) area: $$ \...
0
votes
0answers
17 views

Operator notation for the symmetric and skew-symmetric parts

It is well known that any square matrix can be decomposed as the sum of its skew-symmetric and symmetric parts as $A = A^{skew} + A^{sym}$, where $A^{skew}=\frac{1}{2}(A-A^\top)$ and $A^{sym}=\frac{1}{...
0
votes
0answers
40 views

Getting along with differential of a function, Leibniz notation

$\frac{dx}{dy}$ is one of worst things man can occur in mathematics. This conclusion can be supported by Wikipedia article on it. For example: Augustin-Louis Cauchy (1823) defined the differential ...
1
vote
1answer
52 views

Correct notation for presenting solutions to equations

Let's say I have a cubic equation $(x-a)(x+b)(x-c) = 0$, and I want to represent the solutions to this equation, what is the formal/conventional way that one would arrive and state the solution to the ...
0
votes
0answers
20 views

What does “co supp” mean?

For $S\subset \mathbb{R}^l$, let $\psi:\mathbb{R}^l\to M(S,\mathbb{B}_{S})$, where $M(S,\mathbb{B}_{S})$ is the set of probability measures on $S$ with its Borel $\sigma$-field. For $p\in S$, what ...
2
votes
2answers
50 views

Meaning of $\searrow$ notation

I was reading a paper which contained the phrase "... for any sequence $c_n \searrow c$ there exists $n_0$ such that ..." I am not familiar with this notation, does $\searrow$ have some common ...
6
votes
5answers
167 views

Is $f(x,y) = f(\mathbf{x})$ abuse of notation?

A scalar function $f(x,y)$ is often written as $f(\mathbf{x})$, where $\mathbf{x} = (x,y)$, but as far as I know, there is a difference between the scalar function inputs $(x,y)$ and the vector input $...
0
votes
1answer
49 views

Formal representtion of cases in mathematics

Trivial part: Solving a quadratic equation $ax^2 + bx + c = 0$, where $a,b,c$ are real numbers, requires to consider two cases: (i) $a=0$ and (ii) $a \neq 0$. In the first case there is only one ...
0
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2answers
38 views

Do these infinite expressions have a meaning?

While playing with expressions, I came up to the following "infinite sums". I haven't seen them anywhere else but maybe I didn't look long enough. Find the values of $s$ and $t$. $$s=\sum\nolimits_{1}^...
1
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1answer
17 views

Translating between Set Theoretic and Interval notation

Given an open interval (or closed it makes little difference to the question) on the Real Field $(a, b)$, where $a,b$ are real numbers, and an arbitrary predicate $P(x)$, which is true for all ...
1
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3answers
38 views

What is $\frac{\mathrm{d}f(x,x)}{\mathrm{d}x}$

Related: Is $\frac{\partial}{\partial x} f(x,y(x))$ ambiguous? If $f(x,x) = x^2$, is it correct to say that $\dfrac{\mathrm{d}f(x,x)}{\mathrm{d}x} = 4x$? I know that $\dfrac{\mathrm{d}f(x,y)}{\...
0
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2answers
48 views

Is $\frac{\partial}{\partial x} f(x,y(x))$ ambiguous?

Given $f(x,y) = xy^2$, then clearly $\dfrac{\partial}{\partial x} f(x,y) = y^2$ However, if we set $y^2 = x$, then we get $f(x,y(x)) = x\times x$ By definition $\displaystyle \dfrac{\partial}{\...
0
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0answers
13 views

Is there a notation for space of all random variables?

Is there a common notation for the set of all random variables of the form $X:\Omega\to\mathbb K$? I'd like to use it as a shorthand when i need to specify, e.g., that some function actually maps ...