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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1answer
122 views

Free presentations of $\mathbb{Z}G$-modules

Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free ...
0
votes
1answer
259 views

Definition of a subcomplex of a $\Delta$-complex

I am taking the following as the definition of a $\Delta$-complex. (i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$. (ii) for each $\alpha \in I_n$, one takes a ...
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vote
2answers
153 views

Informal Equivalents of Mathematica “Set” and “SetDelayed”

How would one distinguish between what is meant by Mathematica's "Set" and "SetDelayed" functions in informal mathematical notation? Is there a way to make this distinction any any reasonably standard ...
10
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8answers
10k views

What is the difference between equation and formula?

Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference. ...
4
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1answer
311 views

Equivalent definitions of ordinals?

The first definition of an ordinal number I found was that an ordinal number is the $\in$-image of a well-ordered set $(A,\lt)$. From this definition it was derived that an ordinal is just the set of ...
3
votes
1answer
576 views

What is the formal definition of a one sided limit?

I'm looking for the formal definition of $\displaystyle \lim_{x \to a^+}f(x) = L$ and $\displaystyle\lim_{x \to a^-}g(x) = M$ I took a guess at it intuitively, but I need to make sure this is ...
56
votes
9answers
8k views

Is infinity a number?

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school ...
3
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0answers
336 views

What is the definition of mathematics? [closed]

Is there an exact definition of mathematics ? ...if yes, then what is it ? ...if no then why not ?
4
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1answer
105 views

Help on a definition

In many books I find these two definitions of Ramsey ultrafilters, extremely similar but different: 1)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists ...
5
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4answers
497 views

What differences are between $\mathbb{E}^n$ and $\mathbb{R}^n$

What differences are between the two notations $\mathbb{E}^n$ and $\mathbb{R}^n$? Do they represent/define the same space set with the same structure(s)? Thanks and regards!
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2answers
451 views

Is this Vector operation defined? Does it have a name?

Let's say I have 2 vectors: [a, b, c] [x, y, z] And I need to do an operation like the following for a computer program: ...
6
votes
1answer
874 views

what is f prime?

currently taking Measure and Integration course, which seems to have a different definition of f'. traditionally, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ but in folland's book, it seems ...
3
votes
1answer
169 views

Definition of the union of structures?

Using the logic definition of a structure as a set coupled with finitary functions and relations, what is the definition of the union of two structures $\mathfrak{A}_1 \cup \mathfrak{A}_2$? I ...
11
votes
4answers
1k views

Which is the “proper” definition of a geodesic curve?

I'm taking a course on differential geometry, and up until now I'd always thought that the definition of a geodesic is (loosely speaking) a curve on a surface with the minimal length between its ...
6
votes
3answers
645 views

Why not define 'limits' to include isolated points?

If I understand correctly, most definitions of 'limits' require that the function either a) be defined in an open neighborhood around the relevant point or b) more permissively, that the relevant ...
2
votes
2answers
215 views

Extending “as x approaches a” to “as g(x) approaches a”

All the definitions I can find of a limit (with functions from R to R) define something like: "as x approaches a, f(x) approaches L" Where x is treated as a variable that is quantified over in the ...
1
vote
1answer
235 views

Why are there two different Leibniz notations?

Why do we have dy/dx with the regular d, and 'del y/del x' with the 'funny' d? I can easily find definitions for each expresion, but the definitions appear to be logically equivalent. However, they ...
17
votes
3answers
576 views

Do you know of any Calculus text defining the exponential and logarithm functions in an alternative way?

The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look ...
6
votes
6answers
1k views

Definition of an Ordered Pair

"The ordered pair $(a,b)$ is defined to be the set $\{\{a\},\{a,b\}\}$." ~ Hungerford's Algebra (p.6) I think this is the first time that i've seen this definition. I've read the wiki page. Is it ...
3
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2answers
3k views

Closed Subspaces of Vector Spaces

Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean? Does this mean closed under the norm ...
10
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1answer
4k views

Difference between a theorem and a law

There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra: Theorems Idempotent Involution Theorem of Complementarity Laws Commutative ...
6
votes
1answer
2k views

Relationship between tuples, vectors and column/row matrices

I am taking a course in linear algebra at the moment, and the book I have uses $1\times n$ matrices, $n\times 1$ matrices and $n$-tuples to represent vectors. In condition I have been taught that ...
10
votes
3answers
607 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
8
votes
8answers
391 views

Is the 'variable' in 'let $y=f(x)$' free, bound, or neither?

Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate ...
3
votes
1answer
119 views

does the definition of model depend on a theory or just a signature?

Σ:signature T:Σ-theory For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L ...
11
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2answers
579 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...
4
votes
1answer
296 views

Definition of simplicial approximation

I have been given the following definition of simplicial approximation in lectures: Let $K, L$ be simplicial complexes and $f : |K| \to |L|$ be a continuous map of their polyhedra. A simplicial ...
2
votes
1answer
80 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...
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vote
2answers
275 views

Almost surely in definition of Martingale

$(X_n), n \in \mathbb{N}$ is a stochastic process. I saw in one definition of Martingale that $$E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$$ I understand what "almost ...
5
votes
5answers
3k views

Difference between kernel and function?

I have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time we ...
2
votes
2answers
545 views

Are negative or noninteger powers still power series?

I saw definitions and theorem about power series are in the form of $\sum_{k=0}^n a_k (x-x_0)^k$. And it definitely doesn't include negative or noninteger powers. Nevertheless, I saw the theorems like ...
1
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1answer
3k views

What is a standard curve?

I know you may find definitions out there for standard curve. But, how can we define it in a way that makes it more understandable and clear of what it does?
0
votes
2answers
163 views

The Polynomial concept needs to include both variables and contants?

As in Wikipedia: In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants. So, it's only considered a polynomial if ...
0
votes
2answers
417 views

Proving that a point is a Stationary point with the definition of the derivative

How do you demonstrate that a local minimum of a function has its derivative equal to zero with the definition of the derivative.
2
votes
2answers
180 views

Is everything an expression?

Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions ...
4
votes
2answers
415 views

Describability of a set

As Wikipedia says, there are two ways to describe a particular set: intensional definition and extensional definition. For a set able to be described by extensional definition, is it necessary and ...
15
votes
3answers
568 views

Generalization of a ring?

I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation. Couldn't we continue stringing together ...
2
votes
2answers
498 views

Definition of Symplectic Matrix

In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given: $$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$ is symplectic if and only if: ...
13
votes
3answers
2k views

What is a special function?

When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged ...
5
votes
2answers
363 views

What is a simple loop?

I'm asking here because no textbook or website that I know of gives a definition of the above mentioned term. Since there's no obvious way (that I can think of) to define a normal subloop, I don't see ...
1
vote
1answer
123 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
4
votes
1answer
362 views

Definition of direct image?

I read a little remark in a topology text that the direct image of an open set under a continuous mapping is not necessarily open. What is the definition of direct image in this case? I tried ...
4
votes
1answer
533 views

Definitions for limsup and liminf

I was wondering what are the general spaces that the concepts limsup and liminf can apply to? Is complete lattice one of them? Also How about metric space? What are limsup and liminf specified with ...
6
votes
3answers
1k views

What is the proper geometric description of a the oval used for a horse racetrack?

I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name?
7
votes
4answers
1k views

Alternate definition of prime number

I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works: A prime is a quantity $p$ such that whenever $p$ is a factor of ...
18
votes
7answers
2k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
17
votes
3answers
1k views

Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?

I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...
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6answers
4k views

What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
7
votes
7answers
1k views

Infinite limits

Does a limit that has the value of infinite exist or not? I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts ...
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11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...