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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2
votes
1answer
38 views

Math notation: What does $I$ mean in this context?

Sorry if this is a noobish question, but I don't know what $I$ means in this context: $$\hat{f}(X) = \sum_{m=1}^5 c_m I\{(X_1, X_2) \in R_m\}$$ I am reading about Decision Trees in The Elements of ...
3
votes
4answers
57 views

What is the reason behind the current Order of Operations? (PEMDAS)

After reading through a few other questions I was just asking myself: How was the Order of Operations defined, and why is it this specific order and not a different one? Most of us know things like ...
1
vote
1answer
27 views

Understanding Fréchet derivative and directional derivative

Let $~f: U \subseteq \mathbb R^n \to \mathbb R$ where $U$ is open, then the Fréchet derivative defines the function $f': U \to (\mathbb R^n)'$ that maps vectors to a unique linear functional for which ...
0
votes
0answers
27 views

Confusion regarding a statement in Atiyah-Macdonald

Atiyah-Macdonald says the following: If the ideals $a_i, a_j $ are co prime, then $\Pi a_i= \cap a_i $ What does this even mean? For example, we know that $(2), (3) $ are co prime in the ring ...
0
votes
2answers
19 views

Conditional distribution distributed as notation

What's the proper way to write: $$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$ Some people write it as $X|\mu \sim \mathcal{N}(\mu, 1)$, however I find this confusing as it isn't clear what is a ...
3
votes
2answers
61 views

Are sequences properly denoted as $\subset$ of a set, or $\in$ a set?

Given some sequence $(x_n)$ of some subset $M \subset \mathbb{R^n}$, is it more appropriately to denote $(x_n) \subset M$, or $ (x_n)\in M$? This stems from confusion of using "in" i.e. whenever ...
1
vote
1answer
20 views

notation - sigma algebra

i am trying to study borel sets and sigma algebra- and came across the following on wikipedia (https://en.wikipedia.org/wiki/Sigma-algebra): (However, after n flips of the coin, you may want to ...
3
votes
2answers
57 views

Quick Exponent Clarification

$N = 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5}}}}$ In the following equation is N equal to $5^{5^4}$ or $5^{(5^{(5^{(5^5)})})}$? One of them is huge compared to the ...
2
votes
1answer
48 views

Does using the notation $f(x)$ imply the relation is a function? [closed]

If I have a mapping that is not a function, can I still use $f(x)$ to describe it?
6
votes
1answer
54 views

Are custom named functions acceptable notation?

A custom name being, for example, my function name (MFN): $MFN(x) := ax + b$ As contrasted with: $\delta(x) := ax + b$ Questions: Is it permissible to name the function $MFN$ above? Or is this ...
1
vote
1answer
56 views

What does the notation $U\mathfrak{sl}_2$ mean, and why is the $U$ written in a different typeface to the $\mathfrak{sl}$?

A representation theory homework problem asks me to determine the finite dimensional irreducible representations and the finite dimensional indecomposable representations of $U\mathfrak{sl}_2$. I ...
1
vote
1answer
34 views

Could someone explain the notation of the average of quaternions equation?

The equation has some notation that is difficult to find the meaning for. It is equation (3) in the paper 'Quaternion Averaging' by F. Landis Markley, et al. on page 3 under 'The Average Quaternion'. ...
3
votes
2answers
67 views

What does the symbol $\in$ mean?

I am currently studying discrete mathematics, and I have very little background knowledge, and I was wondering what the symbol "$\in$" means. The following is where I encountered this symbol:
1
vote
1answer
58 views

What does the symbol <<< mean? [duplicate]

Can anyone give the LaTeX code for the unusual symbol <<<, and perhaps provide some good examples of its use? Does it mean "much much less than"?
1
vote
1answer
26 views

Partial vs. complete definition of a function

Suppose I define a function as $f(x,y)=2(x+y)$. Compare that definition to $f:\mathbb{R}^{2}\rightarrow\mathbb{R}, f(x,y)=2(x+y)$, which also gives the (co-)domain. Is there any standard way to refer ...
25
votes
10answers
3k views

What is the accepted syntax for a negative number with an exponent?

A friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the ...
14
votes
4answers
1k views

What is the symbol for primes?

Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well-known, and each one gets its own distinguished symbol. Is there any reason that primes don't get their ...
0
votes
0answers
19 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
0
votes
1answer
73 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
0
votes
0answers
31 views

Uncountable Kronecker Delta?

If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice). Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set ...
2
votes
1answer
70 views

Meaning of “such that”

The use of the term "such that" confuses me I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$. Is there any difference ...
2
votes
2answers
53 views

Does anyone know when I would use this symbol ($\supseteqq$) and meaning?

Does anyone know what this symbol means? Where would one use it? Someone recently asked me but I do not know what it means. I have seen it with just one line underneath to denote subset. With an ...
0
votes
1answer
44 views

Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module ...
2
votes
2answers
149 views

Can someone show me why mathematicians use $d\mu$ instead of $dx$ for Lebesgue Integral over $u(x)$

I am an engineer and I learned my Lebesgue integral from an engineering text which dumbed down a lot of stuff, most prominently all Lebesgue integrals were introduced as $\int_\Omega u(x) dx$ instead ...
2
votes
2answers
27 views

How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
3
votes
2answers
47 views

Notation for union / intersection (in the same way $\pm$ stands for plus / minus) - is this a good idea?

Note: $F$ is a class of sets. I was solving a problem in Apostol's Calculus Volume 1. It is to show that $$B-\bigcup_{A\in F} A=\bigcap_{A\in F}(B-A)\qquad\text{ and }\qquad B-\bigcap_{A\in F} ...
0
votes
0answers
40 views

Formulation: Smooth vs. finitely often differentiable

I treat the cases $f\in C^\infty$ and $f\in C^k$ in different sections of my thesis. While I am happy with the title smooth functions for the first section, I am not so sure if finitely often ...
0
votes
2answers
67 views

How to denote an even number in mathematics? [closed]

I need a sign for an even number (not $a\cdot 2$) in my formula. I tried to google it, but I saw only $2a$. Please tell me if there is a special sign?
1
vote
1answer
55 views

Inserting parentheses to produce different values

Without grouping symbols, the expression $$\verb/2 ⋅ 3 ^ 3 + 4/$$ has a value of $58$. Insert grouping symbols in the expression $\verb/2 ⋅ 3 ^ 3 + 4/$ to produce the indicated values. ...
0
votes
1answer
18 views

Notation for polynomials and equating coefficients

I am reading a paper that defines $P_k(s|t)$ as a polynomial of degree $k$ in $s$ given $t$. Does this mean that each term is of the form $f_{k}(t)s^{k}$? (What does "given $t$" mean?) The paper ...
0
votes
2answers
47 views

What's the meaning of the $R(f(x),g(x))$ in $\int R(f(x),g(x))?$

Usually when I'm reading about integration, there is a notation for integrals on some forms, for example: $$\int R(\sin(x),\cos(x)) \;dx$$ Obviously I've deduced that this represents functions that ...
9
votes
3answers
849 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
2
votes
2answers
38 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
1
vote
1answer
27 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
1
vote
1answer
89 views

Should I use set notation or list notation when writing out a basis of vectors?

I think in Sheldon Axler's Linear Algebra Done Right, he makes a comment about why the technically correct way is to write vectors in lists, such as $(v_1, ... v_n)$, while many books use set ...
2
votes
1answer
44 views

How do I determine the domain and range of the following relations using set builder notation?

I have been given the following relations to find the domain and range of using builder notation. (The blue writing is what I have so far) I am just beginning to learn the whole concept of set ...
0
votes
0answers
56 views

What does $D^n$ refer to?

I'm not sure what object $D^n$ is in the following exercise: "Write down an explicit homeomorphism between $D^n/S^{n-1}$ and $S^n$." Thanks!
0
votes
3answers
173 views

Why do mathematicians use $\oplus$ instead of $+$?

What is the historical reason for using $\oplus$ instead of $+$ to denote operations that are generally thought of as addition? Similarly, why is $\otimes$ used instead of $\times$ (or just $\cdot$) ...
21
votes
12answers
4k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
0
votes
3answers
44 views

What does P|a means?

In the proof for the existence of unlimited prime numbers, i saw the following let n be the number of prime numbers as P1,P2,P3,.......Pn let a = P1P2P3....Pn+1 a > Pn and a is not a prime number a ...
1
vote
3answers
25 views

Set numbering representation

I want to represent proper element of sets. For example, $$ A=\{1,2,3,4\} $$ $$A(2)=2$$ But I wonder that this expression is right. Because I know that the set has not order. How can I do this??
4
votes
2answers
56 views

Notation of an infinite union

Is there any difference between: $$ \bigcup_{n =1}^\infty a_{n} \\ \bigcup_{n \in \mathbb{N}} a_{n} $$ From my understanding they both define an infinite union. Is this correct?
0
votes
1answer
15 views

Operator for comparing an n-tuple

Suppose you have to compare the following two finite ordered list of elements (tuples): $(\psi_{i}, R_{i}, A_{i}, \eta_{i})$ and $(\psi_{i}^{*}, R_{i}, A_{i}, \eta_{i})$ and for instance it turns out ...
1
vote
1answer
9 views

Name for map associated with simplicial complex

Given a simplicial complex $\Delta$, implied by the construction process there are associated maps sending euclidean standard simplices into the simplicial complex $\imath: \Delta^n \to \Delta$. What ...
1
vote
1answer
28 views

Interpretation - exponential loss formula

i stumbled upon the following formula while studying the concept of exponential loss. I try to understand this formula. I read it as follows: The function that minimizes the expected exponential loss ...
0
votes
1answer
34 views

What is usually understood as DOMAIN and CODOMAIN of a Relation

Suppose I have a relation declaration as $R \subseteq A \times B$, such that $A=\{1,2,3,4\}$ and $B=\{10,20,30,40\}$. And suppose that the definition of this relation is $R=\{(1,20),(3,40)\}$ We ...
0
votes
1answer
123 views

Four mathematical notations for fraction $1/999$ and how to show/present they are equal

I need some help for notation. I need to present fractions in four different format and I'd like to get it right. I just take $1/999$ for example, but of course it could be any fraction with positive ...
11
votes
6answers
816 views

Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and ...
0
votes
1answer
25 views

Matrix,Linear algebra,polynomial,finite field,notation

In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define ...
-3
votes
1answer
47 views

What does it mean by a function $ f(x)=\exp(O(|x|^2)) $ for $|x|$ large?

Given $f(x)$ is continuous in $(-\infty,\infty)$ and $ f(x)=\exp(O(|x|^2)) $ for $|x|$ large. Now I have an I expression like ...