Tagged Questions

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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45 views

Definition of sign

The following definition is in my notes with no explanation: $$\operatorname{sgn}(\sigma)=\begin{cases}1,&\text{if }\sigma(p)(x_1,\ldots,x_n)=p(x_1,\ldots,x_n)\\-1,&\text{if ...
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votes
1answer
18 views

Subset notation with the bar crossed

Reading the book 'An Introduction To Continuous Optimization', I ran across the $\subseteq$ notation, but with the little bar crossed over with a small $45^o$ dash - only the bar, not the whole ...
-2
votes
1answer
59 views

What does this even mean?

You might be thinking oh great another stupid question, and you're probably right. Coming from minimal education to math I find it hard to figure out notation, recently I have been trying to learn ...
2
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2answers
36 views

Notation for a “conditional” set

Is there any commonly used short notation for the following? $A_n = \begin{cases} \{a_n\}, & \text{if $n$ is odd} \\ \emptyset, & \text{if $n$ is even} \end{cases} $ I'm looking for ...
2
votes
3answers
59 views

How to show that something is not logically entailed?

I was just thinking about entailment and would like to know if you can show that something is NOT entailed by the premises. I know that to show $A, A → B \vdash B$, I could just provide a proof ...
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3answers
45 views

Where to put the “such that”, given multiple quantifier

Personally, I would put the "such that" (i.e. the symbol $:$ or $|$) behind any quantification. That is given an assertion $A(x,y)$, I'd write $$ \forall x\in X\exists y\in Y:A(x,y)\\ \exists x\in ...
1
vote
1answer
45 views

What is a good way to compactly write that a number is an integer between a and b?

Specifically, I refer to the following set: $$ \left\{ x\in\mathbb{Z}\mid a\leq x\leq b\right\} $$ where $a\in\mathbb{Z}$ and $b\in\mathbb{Z}$ such that $a<b$. Alternatively, this can be written ...
3
votes
1answer
53 views

Multiple integral differential notation

When writing a multiple integral, I have noticed there is sometimes used a shorthand for writing the differential in the integral. For example in $\mathbb{R}^3$ instead of writing $\mathrm{d}x\ ...
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2answers
19 views

How can I express that a n-tuple contains an element at least once?

This is a very simple question, yet I could not find a satisfying answer for it. Consider the set $S =\{a, b, c\}$. To describe the fact that the set $S$ contains $b$, you can write $b \in S$. But ...
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0answers
14 views

Probability distribution vectors over a set $S$

Given a (discrete) set $S$, is there a standard notation for the set of all distribution vectors over $S$? That is a notation for the set $$\{X\in[0,1]^S\mid\sum_{s\in S}X_s = 1\}$$
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1answer
25 views

Is it bad practice to define a matrix in which the entries are sets?

In one of my other questions (which has no answers by the way - I admit it's rather difficult!), I define a matrix in which each entry is a set. Now that I think about it, I wonder if defining a ...
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2answers
15 views

Function Notations for quadratics

Given $T(x) = ax^2 + bx + c$ Find $a,\,b$ and $c$ if $T(0) = -4$, $T(1) = -2$ and $T(2) = 6$ I first made $C$ for when $x = 0,\,-4$. But don't know were to go from here. Any help?
3
votes
2answers
46 views

notation for first and second derivatives of a power series

I have a power series $$\sum_{k=0}^\infty\frac{c_k}{k!}x^k$$ where $c_k$ is an arbitrary $k$-th term of some sequence. Then ...
1
vote
2answers
31 views

Clarification of the notation $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$

I have a question that uses the following notation: the function $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$ is defined by $$f(x)=\frac{2x-3}{x-3}.$$ I understand that the left side ...
3
votes
1answer
52 views

Use of the concept of subgroup vs field extension

Why is it popular to use the idea of subgroups in cases of groups and field extensions in case of fields? In both case one set is the subset of the other along with the restriction of some additional ...
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vote
2answers
23 views

Square root and principal square root confusion

A few months ago I asked a question about the $\pm$ symbol because I was confused about it... I still carry the same confusion (which really bugs me) but I think the real confusion has to do with the ...
1
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2answers
33 views

Is this rigorous notation?

$$\sum_{n=-\infty}^{+\infty} f(x,n)$$ Is it rigorous to write this ? It feels weird to write "$n = -\infty$" ...
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5answers
104 views

Is 1^2^3 = $1^{2^3}$ or $(1^2)^3$ [duplicate]

Caret ^ signs can be used to describe the power of numbers. Is $1$^$2$^$3 = 1^{(2^3)}$ or $(1^2)^3$ How do you calculate it? Do you start with $2^3$ and then do $1^8$ or do you start with $1^2$ ...
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2answers
33 views

Are there definition of percent?

In a school I was taught that percent is the same as 1/100. But I think that definition is not rigorous enough because that would imply for example that $5+4\%=5+4/100=5.04$ but this seems weird. So ...
1
vote
1answer
28 views

quadratic field extensions of $\mathbb{Q}_p$

Today during class we proved that there were exactly three quadratic field extensions of the $p$-adic number field $\mathbb{Q}_p$. To prove this it was stated that it was enough to look at the group ...
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2answers
26 views

Notational difference, functions and mappings, talking about sets and classes

A Function is a set of pairs such that no two pairs have the same first member. My question summarized: What if I want to consider proper classes of pairs? The closest question to mine I could ...
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1answer
24 views

What does $b^*$ mean?

What is this notation, my book explains nothing of it. I've colored it in yellow! I am guessing it stands for $b^{-1}$ or $b^1$?
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vote
1answer
39 views

What are these tick marks after the x, y, and z called?

What are these marks called and what do they stand for? This is for a Affine Transformation.
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votes
1answer
39 views

When to use implies

I often wonder about notation and what is acceptable. I have seen many different ways of linking equations together, sometimes with just $=$ and others with $\iff$ or $\implies$. Now, I know when ...
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2answers
31 views

If I say, $f|_{k}$, what does that mean?

If I say, $f|_{k}$, what does that mean? Sorry to be short on words but I can not find it anywhere on google so maybe someone could explain it and what we typically use it for. I ran into the notation ...
1
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1answer
27 views

Shorter expression of a special conditions

Let $A$ be a set and $B$ a condition (can be either true or false). Is there any shorter description of the expression $$ x = \begin{cases} A & B \\ \emptyset & \text{otherwise} \end{cases} ...
3
votes
3answers
38 views

Notation for non-empty subset [duplicate]

To denote non-empty subsets, I repeatedly find myself writing $A\subset S, A\neq \emptyset$. Is there any established shorthand for this, you know, like $A\subset S$ can be seen as a shorthand for ...
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0answers
14 views

Notation for discrete cross-correlation

Consider two real valued vectors $x$ and $y$. Suppose $x$ is $m$ dimensional and $y$ is $n$ dimensional with $n \ge m$. What is good notation for the function which returns an $n-m+1$-dimensional ...
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11answers
5k views

Dividing by 2 numbers at once, what is the answer?

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20. Edit: Thanks for your ...
1
vote
1answer
55 views

What is meant by $ab$ on words $a$ and $b$ in $\{ab\ |\ a,b \in Σ^*\}$?

Given language $L$ := $\{ab\ |\ a,b \in Σ^*\}$, $Σ := \{blue, green\}$. Is the notation "$ab$" above taken to be word concatenation, such that $\{bluegreen\} \subset L$? What occurs when $L$ := ...
2
votes
1answer
30 views

“Evaluated at” or “at” notation

Normally a variable that is a function another variable would be represented as in the following fashion: $ V(t) $ (voltage as a function of time). However, my engineering professor (who also wrote ...
2
votes
6answers
52 views

Notation for sum of products

Is there a summation notation for the sum of products made two by two? I have the following expression: $$x_1x_2+x_1x_3+\dots+x_1x_n+x_2x_3+x_2x_4+x_2x_5+\dots+x_2x_n+\dots+x_{n-1}x_n$$
0
votes
0answers
67 views

Help in simplifying this double summation

Can I express the following double summation $$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$ where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,i \in [1:n], j \in [1:m]\}$ while $G(.)$ is any function ...
2
votes
2answers
29 views

What is word reversal $w^R$?

In the following context, what is the formal meaning of "reversal of word $w$"? The free monoid on $A$ is the syntactic monoid of the language $\{ ww^R\ |\ w \in A^*\}$, where $w^R$ denotes the ...
10
votes
0answers
123 views

Why is $J$ sometimes used to denote $\mathbb{Z}_{>0}$?

In older books, such as Rudin's Principles of Mathematical Analysis and Herstein's Topics in Algebra, I've noticed that authors tended to use $J$ to denote $\mathbb{Z}_{>0}$. Does anyone have any ...
1
vote
2answers
22 views

Reason for defining a quantity with “inf”

In some applications (in my case statistics) I find quantities defined using "inf", e.g. $ ABC = \inf\{x|F_X(x)\ge\alpha\}$ Why not define simply: $F_X(x=ABC) = \alpha$ I imagine it has something ...
0
votes
1answer
21 views

Mapping of elements notation - Cohn - Classic Algebra Page 13

So Cohn uses the notation that many have wanted to change to, being $xfg$ rather than $g(f(x))$, and I have had the example: Let $f,g: \mathbb{N} \to \mathbb{N}$, be given by $xf = x + 1,xg=x^2$, ...
1
vote
2answers
24 views

is there a notation to designate the induced homomorphism including base point?

Let $X,Y$ be a topological spaces. Let $f:X\rightarrow Y$ be a continuous function. Fix $x_0\in X$. Define $f_*([r])=[f\circ r]$ for every loop $r$ at $x_0$. Then, $f_*:\pi_1(X,x_0) \rightarrow ...
0
votes
0answers
27 views

Probability and calculus notation

I just need help to make sense of some notation that I have seen on a document related to monte carlo integration. There is a portion talking about expected values of a continuous random variable ...
1
vote
1answer
29 views

What does the notation $\equiv 1\ (\text{mod}\ p)$ mean?

I'm trying to understand the Fermat theory : $a^{p-1} \equiv 1\ (\text{mod}\ p)$ I know that $a\ (\text{mod}\ p)$ gives the remainder of division of $a$ by $p$. So what is $\equiv 1\ (\text{mod}\ ...
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vote
0answers
24 views

Operator for scaling a function?

Let $\mathbb{F}$ denote the set of functions of the form $f: \mathbb{R} \to \mathbb{R}$. I am interested to know whether there exists a well-known linear map $T_\alpha: \mathbb{F} \to \mathbb{F}$ ...
1
vote
0answers
24 views

Product notation $\prod$ when product does not commute [duplicate]

This is kind of a dubious question, but is the product notation $\prod$ often used in noncommutative rings? For example, if $M_i$ are matrices, I guess the common definition of $\prod$ is $$\prod_i ...
2
votes
0answers
28 views

Better notation for a product

Let $A,B$ are two positive integers. Assuming that we have a product of the form $$ \prod_{\substack{a\mid A \\ \gcd(a,B)=1}}f(a). $$ Is there a better notation to be used instead of $a\mid A$ and ...
2
votes
1answer
29 views

What does the following set symbol notation mean

[6] x [6] -> Z I know it's the cartesian product of [6], but I don't quite understand what [6] means? Does it mean all numbers until 6, or is another way to write {6}?
2
votes
1answer
17 views

Incongruencies with derivatives and differencials

I read in Piskunov that the increment $\Delta y$ of a function can be written as: $\Delta y = f'(x) \Delta x + \alpha \Delta x$ And, when ${\Delta x\to 0}$ , $dy=f'(x)dx$ The problem is, doesn't ...
1
vote
1answer
47 views

When is Leibniz' notation for derivatives useful?

So Lagrange's $y'$ and Leibniz' $\frac{d}{dx}y$ seems to be the two most common notations for differentiation, but it seems puzzling to me that there are two notations for this. I've been taught ...
0
votes
0answers
17 views

Does the diagonal of a gradient exist?

If $\nabla \cdot \vec x$ is okay, why not $\text{diag}(\nabla)\vec x$? Should we write $\nabla \vec x$ or $\nabla \vec x^\intercal$ since the result is an outer product? E.g., if I want to write ...
0
votes
2answers
16 views

Notation for the “scalarization” of a vector with a single non-zero entry

Suppose I have a vector $v$ in the complex space $\mathbb{C}^N$ with only a single non-zero element. Is there a standard notation to replace the vector with a scalar equal to the non-zero value of ...
0
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1answer
43 views

Understanding a weird notation when proving a theorem

I'm reading a paper that's trying to prove a theorem. However there is a weird notation that I couldn't understand. First they present the theorem and then they present two claims. In the first claim ...
1
vote
0answers
48 views

What does $V^*$ represent in linear algebra?

If $V$ is a vector space, then what does the notation $V^*$ normally stand for? Thank-you