0
votes
1answer
42 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
3
votes
0answers
64 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
1
vote
2answers
55 views

Abbreviate a tuple with a variable?

In a book I found the following notation $M = \{(x_1,y_1,z_1), \ldots, (x_n,y_n,z_n)\}$ for a set of 3-tuples. The author always refers to a tuple by writing $(x_i,y_i,z_i) \in M$. I'm wondering if I ...
0
votes
0answers
30 views

Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
2
votes
1answer
25 views

Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
0
votes
3answers
66 views

Set question - $ ℤ^+ = ℕ$ [duplicate]

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$ if not, why? Thank you in advance! I appreciate your help!
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
3
votes
2answers
47 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 $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
1
vote
4answers
81 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
1
vote
1answer
18 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
1
vote
2answers
40 views

Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
2
votes
1answer
22 views

What is the name for the property that a subset of a set follows the same rules as the set?

I have a set that follows a certain property and I want to say that the subsets of this set also follows the property. What is this called? I know that closure under an operation means that performing ...
0
votes
1answer
26 views

A notation question on how to properly denote a function that takes inputs only of a certain form.

Suppose I have a set $B = \{n^2 + n + 1 : n \in \mathbf{N}\}$ and I want to define a function $g: B \rightarrow \mathbf{N}$ that only accepts as it's arguments numbers of the form $n^2 + n + 1$ for $n ...
1
vote
2answers
167 views

What does the 'triple line equals' sign with a strikethrough mean?

I understand that the triple equal sign can mean "identical to" or "defined to be equal to", so intuitively, I assume that the triple equal sign with a strikethrough means "is not identical to" or ...
1
vote
1answer
45 views

Big-$O$ notation definition.

I have come across the notation $$...+\ O_R(x)$$ where $R$ is any positive real number, and $R<2$. What is the term big $O_R$ of $x$ referring to?
1
vote
1answer
77 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
0
votes
1answer
179 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
1
vote
2answers
87 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
1
vote
1answer
36 views

Notation of double-sided infinite sum

The notation $\sum_{k=1}^\infty a_k$ always means $$\lim_{n\rightarrow\infty}\sum_{k=1}^n a_k.$$ What about $\sum_{k=-\infty}^\infty a_k$, such as in the Laurent series? Does it always means ...
1
vote
2answers
109 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
3
votes
2answers
64 views

degree of commutativity

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?
1
vote
2answers
56 views

“Preimage” of a binary relation

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$? ProofWiki calls $X'$ the preimage of $R$, denoted as ...
0
votes
2answers
104 views

Meaning of NOT all but finitely often

Can someone clarify for me the meaning of the statement "NOT all but finitely often"? It's driving me crazy. I'm not able to break it up. Thanks.
0
votes
1answer
119 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
-3
votes
1answer
86 views

What is this symbol?

This symbol: I found on the Wikipedia page for the Law of Cosines. It looks kind of like a lowercase Gamma (γ), but it doesn't specify so I'll wait to see what you guys think. Thanks
5
votes
1answer
82 views

The use of = vs := for definitions

I have seen the following conventions, e.g. We define $K=\mathbb C$ ... We define $K$ to be $\mathbb C$ ... We define $K:=\mathbb C$ ... I prefer number 3, because it is concise and it is clear ...
1
vote
1answer
99 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
0
votes
1answer
98 views

question about Graph Theory notation

I'm just starting to learn graph theory. I have two questions about notation: 1). For a graph $G$ we denote the vertex set $V$ and the edge set $E$ by $G=(V,E)$. So we have a graph $G=$ ...
0
votes
1answer
82 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
3
votes
2answers
84 views

Conventions on definitional if(f)

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance: Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such ...
4
votes
4answers
282 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...
17
votes
12answers
3k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
2
votes
1answer
107 views

how to write the process of decomposition of a graph into shortest closed sub graphs

If I want to decompose a graph in to possible shortest closed cycles (as shown in right side). then how can i describe this process with mathematical notations. to understand please refer below ...
0
votes
1answer
228 views

Mathematical notation of graph subdivision

If anyone can define a directed graph subdivision with mathematical notation, please post a response. My second question is: Irrespective from the planar embedded graph or not, is this definition ...
2
votes
2answers
330 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
3
votes
0answers
66 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
1
vote
2answers
165 views

What is the meaning of the expression $\liminf f_n$?

I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
2
votes
1answer
72 views

“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
1
vote
3answers
150 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
2
votes
2answers
127 views

A simple notation question on grad and Lp norm

What does this notation $||\nabla u|| _p$ mean? I would like an exact definition. Also I have seen $|\nabla u|_1$. Does this mean the $\sum_i|\partial_i u|$?
2
votes
0answers
86 views

Where can I find a description of math language symbols?

I am reading math articles. I meet math symbols. For example $\exists$ or $\forall$. For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$ Where can I find full ...
0
votes
2answers
77 views

Exponentation vs Power

What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?
5
votes
4answers
292 views

What is the correct definition of the absolute value of $x$, $|x|$?

What is the correct definition of the absolute value of $x$, $|x|$? Option A $$ |x|= \begin{cases} -x&\text{if } x < 0\\ 0& \text{if } x=0\\ x&\text{if } x>0 \end{cases} $$ Option ...
7
votes
1answer
139 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
10
votes
3answers
467 views

What is the purpose of the $\mp$ symbol in mathematical usage?

Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please ...
1
vote
5answers
2k views

What does it mean when a function is finite?

When someone says a real valued function $f(x)$ on $\mathbb{R}$ is finite, does it mean that $|f(x)| \leq M$ for all $x \in \mathbb{R}$ with some $M$ independent of $x$?
6
votes
1answer
406 views

What do these arrows mean? (Froda's Thm)

I was reading "A Course in Probability Theory" by Kai Lai Chung, and in the book he was discussing discontinuity of monotonic functions, and after doing some searching online to learn more about the ...
1
vote
1answer
176 views

What is the nature of the definition symbol?

A question about definitions and also notation, illustrated with a trivial example: Let $a,b,c\in\mathbb{N}$ and $$a:=2,$$ and $$b:=1$$ I postulate the the following formula holds ...
3
votes
1answer
117 views

What do mathematicians call the Two's Complement on 8-bits group?

It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$ What is an elegant ...
3
votes
2answers
998 views

$\{0,1\}^n$ and $[0,1]^n$ notations

Can someone please help me clarify the notations/definitions below: Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s? Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector ...