Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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2
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1answer
24 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
2
votes
3answers
28 views

Can someone explain me how to read members of a set to prove uncountability?

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all ...
0
votes
0answers
43 views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
2
votes
1answer
31 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
3
votes
1answer
39 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
0
votes
2answers
50 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that ...
4
votes
3answers
148 views

What precisely is a vacuous truth?

Is there a proper and precise definition that goes something like this? Definition. A statement $S$ is a vacuous truth if ... ...
0
votes
1answer
26 views

$\epsilon$ rules for uniform and pointwise convergence.

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but ...
2
votes
3answers
65 views

Can I create my own function like Trigonometric or Exponential

When I want to solve mathematical problems, most of the time I meet the following functions Algebraic like polynomials. Trigonometric like sin(), cos(), tan(), cot(). Logarithmic like log(). ...
0
votes
2answers
78 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
3
votes
1answer
25 views

Linear function definition

I'm trying to figure out what is linearity and what is a linear function. But the wikipedia page confused me. Firstly it defines as polynomial : $f(x) = ax +c$ Than it defines as linear map: ...
1
vote
4answers
59 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 ...
6
votes
4answers
246 views

What is the *exact* definition of a bounded subset in a metric space (in relation with the Heine-Borel Theorem)?

I see quite a lot of different definitions of a bounded space. For instance, from nLab: Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that ...
1
vote
1answer
60 views

A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
0
votes
4answers
83 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
2
votes
1answer
28 views

Clear definition of degeneracy of a graph.

There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either. Could someone please clarify is the degeneracy of a graph $G$ ...
1
vote
1answer
50 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
0
votes
2answers
33 views

Confused about class definition

I find this in my Set theory material: [0] = {x:0==x(mod2)} = {x:2|0-x} , where I'm replacing equivalence sign ("=" with extra horizontal line) with double ...
5
votes
0answers
64 views

Are the words “function”, “map”, and “mapping” synonymous?

Is it correct to say that "A function or a map or a mapping is a binary relation such that ..."
3
votes
2answers
48 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
1
vote
1answer
30 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
0
votes
0answers
25 views

About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in ...
0
votes
1answer
21 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
1
vote
2answers
45 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
1
vote
1answer
14 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 ...
0
votes
4answers
40 views

Definition of Normalized Number

Which is correct? Are they both correct? Definition 1 A floating point number is called normalized if the leading digit of the fraction is nonzero. for example $(0.10101)_{2}\times 2^{3}$ is ...
0
votes
2answers
40 views

What is the easiest way to prove by induction?

Is there any easy way to do this? I get the basic step.. where you prove it for some number.. but I don't get the induction step. Do you literally take the given equation that you just proved with ...
10
votes
3answers
393 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
0
votes
2answers
35 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
0
votes
1answer
87 views

Is the value of the sum $1-1+1-1+1-1+\cdots$ does not exist? [duplicate]

Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer. And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ ...
1
vote
2answers
36 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 ...
1
vote
1answer
24 views

Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair

Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b ...
0
votes
0answers
21 views

Depth in acyclic graphs

I am struggling to understand a definition in a paper: Given a acyclic (directed!) graph $D=(V,E)$ we define a sequence $Q_i \subset V(D)$ of sets: $$Q_0 = \emptyset,$$ $$ Q_i \textrm{ is ...
1
vote
0answers
31 views

Solvable Group, which Quotients need to be Abelian?

In Wikipedia it says a group $G$is solvable if it has a subnormal series $\{e\}=G_1\lhd G_2,\dots \lhd G_n=G$ where $G_i$ is a normal subgroup of $G_{i+1}$ and all the factor groups are abelian. My ...
1
vote
2answers
49 views

How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
3
votes
1answer
25 views

which is the usual definition of Argument?

Let $z\in\mathbb{C}\setminus\{0\}$ Then, there exists a unique $\theta \in [0,2\pi)$ and $\phi\in(-\pi,\pi]$ such that $z=|z|e^{i\theta}=|z|e^{i\phi}$. Between $\phi$ and $\theta$, which is the ...
0
votes
0answers
14 views

In the definition of NP, is it required to have polynomially bounded length of certificate?

So given the definition in our lectures, we were told that NP is defined as the set of languages $L$ s.t. there exist a polynomial time bounded Turing-acceptor M s.t. $L ={w: M accepts(w#c) for some ...
0
votes
0answers
37 views

Particular “upper bound” and “lower bound” of sequence

Is it possible to define the following? Let $a:\Bbb{N} \to \Bbb{R}$, and $M \in \Bbb{R}$, $M$ is ? if $$\exists n \in \Bbb{N}(\forall p \in \Bbb{N}(p\geq n \to a(p)\leq M))$$ Let $a:\Bbb{N} \to ...
2
votes
2answers
24 views

Does the definition of “mean”/“average” require the result to be in the domain set?

If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean ...
0
votes
1answer
46 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
1
vote
2answers
61 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
0
votes
1answer
23 views

Question about measure on set that is not in $\sigma$-algebra

I think I have problem with badly written book, or I just can't understand statement. Let $(X,\mathcal{A},\mu)$ be any measure space and $\mu^*$ be outer and $\mu_*$ inner measure inner measure ...
2
votes
2answers
78 views

Geometrically reduced variety

What is a geometrically reduced variety (or geometrically reduced algebraic set if you will, a variety has not been assumed to be irreducible in this definition)? I tried looking up on the internet as ...
1
vote
1answer
38 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
0
votes
2answers
61 views

I Need Help Understanding the Formal Definition of A Limit

I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that: $$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ ...
3
votes
2answers
75 views

Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
1
vote
0answers
18 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
5
votes
1answer
49 views

Is there a rigorous definition of the term “coordinate system”?

You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no ...
1
vote
1answer
68 views

Is this a correct definition of the natural numbers in ZF?

Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$ $y\in{s}\rightarrow(y$ is transitive$)$, and if $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is ...
2
votes
0answers
36 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...