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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0
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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
34 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
67 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
50 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
63 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}{\...
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1answer
33 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
39 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
votes
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
votes
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
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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
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0answers
52 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
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2answers
75 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
35 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
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1answer
57 views

What does “pcr” stand for?

For example, as shown in this OEIS page: $P(n,4)=\frac{2n^3+6n^2-9n-13+(9n+9)\text{pcr}\{1,-1\}(2,n)-32\text{pcr}\{1,-1,0\}(3,n)-36\text{pcr}\{1,0,-1,0\}(4,n)}{288}$
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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
41 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^{...
2
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3answers
32 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
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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
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1answer
58 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
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0answers
18 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
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0answers
41 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
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1answer
54 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
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0answers
22 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
177 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
50 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
45 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 ...
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3answers
39 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
votes
2answers
52 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
votes
0answers
14 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 ...
0
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0answers
18 views

Total derivative using $(x,y)\mapsto f(x,y)$ notation

Related to What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$? Is it possible to make sense of $(x,y)\mapsto f(x,y)$ when taking the total derivative ...
12
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5answers
294 views

What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$?

Is there not any difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$ as long as your function has one variable? $f(x) = x^3\implies \left\{\begin{align}&\dfrac{\...
1
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0answers
26 views

Understanding Matrix and Vector Notation

I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem). Problem: For equation (2), I think it should be $\...
1
vote
1answer
34 views

What is the proper notation for vectors that uses only integers? $\Bbb Z^n$?

How do I denote vectors, similar to $\Bbb R^n$ but with integers instead of the reals? Just $\Bbb Z^n$? Would vectors of exclusively positive integers be $\Bbb Z^{{+n}}$ then? (since $\Bbb Z^+$)
1
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0answers
87 views

Math notation to show two numbers in a range that added together get the max of the range [closed]

I am completely new to math notations, it's been about 30 years since high school, and I am writing a research paper (completely on my own, not for a degree). I basically want to show that two real ...
-1
votes
1answer
49 views

The $n=0$ term in the power series $\sum_{n=0}^\infty a_n x^n$

This question is about the definition and notation for the $0^{th}$ term of the power series: $$\sum_{n=0}^{\infty} a_n x^n$$ There are two possible ways to interpret this term: 1) It is just a ...
1
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0answers
21 views

Understanding the Degree Distribution of Watts-Strogatz Model

In the Watts-Strogatz model, the degree distribution is given as: $P(k) = \sum\limits_{n=0}^{\min\left\{k-\frac{K}{2},\frac{K}{2}\right\}} C^n_{\frac{K}{2}} (1-p)^{n}p^{\frac{K}{2}-n} \frac{\left(\...
0
votes
1answer
48 views

Clarification of the topology lemma “Any continuous and open injection of the open disk extends over the circle”

My elementary topology 1 class last semester used the book "Topology: Point-Set and Geometric" by Paul Schick, and covered through the end of chapter 8. I am working through the rest of the book on ...
2
votes
1answer
24 views

What does $ K^{\alpha}$ mean?

This is in context of a statement in galois theory: If $F \subseteq K \subseteq L$ and $K$ is splitting over $F$, then $K^{\alpha}=K$ for each $\alpha \in Aut(L)$.
0
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3answers
146 views

How to suppress the words “if” and “then”?

My math teacher keeps making us write mathematical sentences with "regular" words. I always ask her if it is possible to supress them but she always says "no" or she starts laughing. Take for example ...
2
votes
1answer
27 views

Symbol to show that implication is one-sided

Sometimes A => B is true, but B => A is not true and this fact is important and not obvious. Is there a short symbol to write it in order not to write "A => B and not B => A"
1
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1answer
49 views

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Let the $3x+1$ map $$ f(n) = \begin{cases} 3n+1 & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} .$$ Now we read the Wikipedia's page for the Collatz ...
0
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0answers
36 views

What does $F[x]$ mean?

Lemma: $F$ is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ mean?
2
votes
2answers
122 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
3
votes
0answers
35 views

Is there a math notation/ term for “$f(x_n) \to 0$ iff $g(x_n) \to 0$”?

I have two real-valued functions $f,g$ defined over the $N$-dimension Real Euclidean space: $$ f,g: \mathbb{R}^N\to\mathbb{R}. $$ They satisfy this property: $$ \forall x_n \in \mathbb{R}^N: f(x_n)\to ...
1
vote
2answers
55 views

Is there a name for the logical scenario where A does not necessarily imply B, but B implies A?

A real life example of this is the 'Active' status on Facebook Messenger. (For those interested see this article here, and some Quora answers here for details.) When you are actively using Facebook ...
2
votes
2answers
25 views

Describing sets from running indices

For a paper I have a set of particles which I usually reference by $ p $. The paper is physics related, so I haven't made any real formal definition of the set, and always just relate to them as "the ...