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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4
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82 views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
1
vote
1answer
39 views

Meaning symbol $\overline{\lim_{n\rightarrow \infty}}$ [duplicate]

I found this symbol on a math book that I'm studying. It had never happened before. What does this mean? $$\overline{\lim_{n\rightarrow \infty}}$$
0
votes
4answers
40 views

“$((A\times B) \to C)$” denotes what?

I'm having some trouble understanding notation. The question is For any three sets $A,B,C$ , $((A\times B) \to C) =_c (A \to (B\to C))$ Exactly what does "$((A\times B) \to C)$" denote? Is ...
4
votes
2answers
61 views

What does this symbol mean? (looks like two curves — subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
0
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1answer
19 views

how to write it using sigma notation

I'd like to know how to create a pattern for these sum: 1) $a_{1}a_{2}a_3+a_1a_2a_4+...+a_1a_{n-1}a_n+a_2a_3a_4+a_2a_3a_5+...+ a_2a_{n-1}a_{n}+ ...+a_{n-2}a_{n-1}a_n = ?$ 2) ...
4
votes
2answers
722 views

What does the symbol described below mean? (looks like: sideway u with a line through the middle)

The symbol kind of looks like this: ε, but it's more like a sideways u with a line through the middle.
4
votes
3answers
297 views

What does this combination of symbols mean? $\exists !$

I just want to know what this combination of symbols means: $\exists !$ I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" ...
3
votes
1answer
210 views

What does the following symbol mean? (direct sum? o-plus? — subject: matrix theory)

In this paper equation 11, the author uses a symbol that is a cross in a circle. I believe I have seen that referred to as a direct sum, but I am not completely sure what that is. $$\bigoplus$$ ...
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1answer
1k views

What does this symbol mean in math? ($Z_+$ and $Z_{++}$)

What is the difference between $Z_{+}$ and $Z_{++}$? Is the first all the non-negative integers and the second all the positive integers?
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2answers
647 views

What does this symbol mean? (looks like a lower-right corner — subject: manifolds theory)

Below I have posted an excerpt of Lee's Book "Introduction to Smooth Manifolds" (page 371). I don't know what the symbol means that looks like a lower-right corner, and I cannot find it via the index, ...
6
votes
1answer
112 views

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
0
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0answers
30 views

if the role of a numeral system is to provide a mathematical notation for representing numbers. Then how do notation less numbers look like?

Our numeral system maps a unique(or at least standard) notation to every number in a set. for example : In decimal system we map natural number set by ${1,2,3,4,5,6,...}$. My question is What ...
0
votes
1answer
22 views

Why notations of numbers in standard positional numeral systems when evaluated, evaluates to decimal notation?

For a positional numeral system of radix b, evaluating a notation $a_3 a_2 a_1 a_0$ as shown below results in decimal notation of number, $a_3 a_2 a_1 a_0 = a_3 \times b^3 + a_2 \times b^2 + a_1 ...
0
votes
3answers
32 views

What does $f: 2^{\mathcal{S}}\rightarrow\,\mathbb{R}$ mean?

A function $f: \mathcal{S}^n\rightarrow\,\mathbb{R}$ This is I understand. $x\in\mathrm{dom}\,f$ means that $x$ is a vector of size $n$ where its elements are taken from the set $\mathcal{S}$. ...
0
votes
1answer
15 views

Notation of a linear inequality system.

Sorry to bother with this rather trivial question, but nowhere in my lectures or books can I quite find out what the topmost line means. Maybe I'm forgetting something. Anyway: Line 2 and 3 are ...
10
votes
4answers
698 views

What was the notation for functions before Euler?

According to the Wikipedia article, [Euler] introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical ...
0
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1answer
10 views

Expression as argument in function definition

When a function definition has an expression (instead of just a single variable) as the argument to the function, what does this mean? For example, I have this question (part b): Given a certain ...
7
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1answer
86 views

First usage of the symbol ∈

Concerning a book [1] I am reading the symbol $\in$ was first used by Giuseppe Peano and is the first letter $\epsilon$ (epsilon) of the word ἐστί (means "is"). Does anyone know in which work of Peano ...
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0answers
63 views

The fundamental theorem of calculus: $\frac{\mathrm d}{\mathrm{d}x} \int_a^x f(t)\:\mathrm{d}t = f(x)$ and nothing more

My textbook writes, just as wikipedia, but I'd like to write it shorter. Can the fundamental theorem of calculus be written as $$\frac{\mathrm d}{\mathrm{d}x} \int_a^x f(t)\: \mathrm{d}t = f(x)$$
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vote
2answers
27 views

How to represent the edges of a complete graph?

Let say I have a graph $\mathcal{G}$. Denote this graph by $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. My question is ...
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3answers
26 views

The Notation for Derivatives

"The derivative of a sum is the sum of derivatives" Above theorem can be mathematically expressed as: $$h'(x)=f'(x)+g'(x)$$ where $f(x)$ and $g(x)$ are two differentiable functions. What is the ...
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1answer
48 views
+50

$C^\omega$ notation for real analytic functions

I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used ...
8
votes
1answer
238 views

Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.

Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
2
votes
2answers
39 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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0answers
12 views

Representing trees in Set builder notation?

Is there a way to represent graphs and minimum spanning trees using set builder notation? e.g. I have a weighted graph of n nodes, all connected to each other in a mesh network manner. I am to ...
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2answers
230 views

Addition of Sets which isn't union

today a student asked me to prove $${A} \cup B \cup C = A+ B+ C- A\cap B - A\cap C$$ I really had no idea what precisely the "+" sign meant, they insisted, "You know you just add the sets together"; ...
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votes
1answer
33 views

Set notation of $S^1 \times S^1$

This is a simple question, but should this be written as: $\hspace{120pt}S^1 \times S^1 = \{(z_1,z_2)\in\mathbb{C}\times\mathbb{C}:|z_1|=|z_2|=1\}$
3
votes
1answer
77 views

What are the differences in mathematical notation around the world?

I just learned that $\text{sen}\,x$ is the Portuguese notation for $\sin x$, and I was inspired to ask: What differences are there in how mathematics is written around the world? Note 1: I am likely ...
2
votes
1answer
46 views

Lie derivatives and covariant derivatives (notation)

I am having troubling interpreting a particular expression in differential geometry. It arose in computing the Lie derivative along a unit normal, $n$, of the extrinsic curvature of a sub-manifold ...
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1answer
22 views

Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then ...
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1answer
43 views

Notation question A $\subset \subset B$

I am a bit confused about the notation A $\subset \subset B$ used in functional analysis. The definition I have says: $A \subset \subset B$ iff $A \subseteq B$ and $\bar{A}$ compact in $B$. Wikipedia ...
4
votes
1answer
41 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
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0answers
25 views

Abstract algebra notation

I am new to learning abstract algebra and using multiple books but the notation varies enough to throw me off. Could someone explain to me the differences between: $\mathbb{Z}\left\{ p\right\}$ ...
4
votes
1answer
60 views

Is there a difference between writing $f: X\rightarrow Y$ and writing $f:X\mapsto Y$?

I think I've heard about a year ago that "$\mapsto$" is only used for a bijection, or do they mean the same thing?
0
votes
1answer
41 views

How does the presenter in this video derive this formula?

I am watching this coursera video on entropy (in the information theory sense of the word). Right around the two minute mark the presenter shows two forms for H(p). The first (after the equals sign) ...
0
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1answer
4k views

What does the right arrow mean in the example provided.

My second question here. Does it show that I know very little about mathematics? :) I'm doing Project Euler Question 14 and would like to know what the right arrow → means in: ...
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vote
2answers
28 views

Looking for notation of set of all entries of some matrix?

I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this? Looking for something like ...
1
vote
1answer
73 views

Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
0
votes
1answer
24 views

Notation Question about Rings

If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?
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vote
2answers
35 views

How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$ ( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ ) $$
1
vote
0answers
21 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
14
votes
2answers
318 views

The double factorial notation

The double factorial is defined as $$n!! = \begin{cases} n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1 = \dfrac{(n+1)!}{2^{(n+1)/2}((n+1)/2)!} & \text{ If $n \in \mathbb{Z}^+$, is odd}\\ n \cdot ...
2
votes
1answer
226 views

Notationally, what is the difference between $\Pr(X = x)$ and $P(X = x)$? When should I use each?

I'm talking specifically about probability theory. I was reading some stuff about probabilistic graphical models, and they kept switching the notation in this book, but I couldn't discern the ...
0
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1answer
30 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
2
votes
1answer
70 views

What does $\in$ mean?

I'm reading a textbook on complex analysis and I've come across notation using this ($\in$) symbol. In the context of "an argument of $z = x + iy$ is a number $\phi \in \mathbb R$ such that $x = ...
1
vote
1answer
435 views

Bar symbol over a matrix

So I am reading a paper (not online) and I come across a definition: $$\mathbb E=R\bar R$$ Where R is a complex matrix. I am thinking that it means complex conjugate, but I honestly have never seen ...
0
votes
1answer
17 views

Notation query: matrix projection

I've used the following notation in a report: $G\boldsymbol{w}\in\mathcal{H}_2\,\forall\boldsymbol{w}\in\mathcal{H}_2 $ In other words $G$ projects $\boldsymbol{w}$ back into the same set. But this ...
3
votes
6answers
188 views

What is the shortest way to write the number $1234567890$?

Here's a challenge : find the shortest way to write the number $1234567890$ . There is several ways to write the number $1234567890$ : $1.23456789 × 10^9$ $2×3^2×5×3607×3803$ $617283945×2$ But ...
0
votes
1answer
23 views

Understanding $Po(np)\{A\}$ probability notation

I am trying to read a textbook on probability and am already stuck on what must be basic notation. It says Thus, for example, if $A$ is any subset of $\mathbb{Z}^+$, it follows that for some ...
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6answers
412 views

The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working. $$\begin{align*} u&=f(x)\\ \Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\ \Rightarrow ...