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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5
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2answers
702 views

Precise definition of “weaker” and “stronger”?

If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$? (Or something else?) I feel like I have seen both usages in literature, which is confusing. ...
6
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5answers
540 views

Definition of Ring

I'm studying Abstract Algebra right now, currently covering rings. In the introduction of the subject, I am curious as to why there is no need for there to be a multiplicative identity. I understand ...
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3answers
102 views

Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
0
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2answers
225 views

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree?

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree? and just curious, how many roots does a polynomial with 2 and 1/2 degree have and how ...
1
vote
2answers
143 views

exterior product definition

i have question from vector mathematics,i know that if there is given two vector, for instance $a=\{a_1,a_2,a_3\}$,$b={b_1,b_2,b_3}$; then so called exterior product is determined as $a\wedge ...
1
vote
1answer
209 views

Expressing P = NP as a first order formula

I want to express P = NP in a completely formal way. My first try: There exists an algorithm A and a polynomial bound p such that for all input i, A(i) = true iff i is a satisfiable formula and ...
14
votes
5answers
622 views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
7
votes
3answers
766 views

Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
4
votes
2answers
414 views

True, false, or meaningless?

Are the following two assertions always true, always false or meaningless? $\exists i \in \emptyset$ $\forall i \in \emptyset$ Because it seems that one encounters expressions of this kind fairly ...
7
votes
3answers
665 views

What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?
4
votes
3answers
636 views

Definition of Ring Vs Rng

When I took abstract algebra I learned that a ring was a set that is an abelian group under addition and monoid under multiplication (along with the distributive property). In preperation to tutor ...
3
votes
1answer
1k views

What is Hermite data?

Using fairly simple language, what is Hermite data? I encountered it here, http://www.frankpetterson.com/publications/dualcontour/dualcontour.pdf and could not get an answer on standard StackExchange, ...
4
votes
1answer
358 views

What is this R-like symbol power 2?

I found this in a computer medical research text. What is the meaning of this R-like letter? S, in this context is an iso-intensity surface. [edit] Since context is not sufficient, I think it is ...
2
votes
3answers
194 views

“Defined as” versus “Equivalent to”

This is a lazy question, but very often textbooks use the "$\equiv$" (equivalent to) sign and the "$:=$" (defined as) sign in the same places from book to book. I suppose equivalence to a previously ...
2
votes
1answer
276 views

Set-theoretical definitions of the notion of “structure”

What general set-theoretical definitions of the notion of "structure" are there? By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that ...
19
votes
3answers
2k views

Is a line parallel with itself?

Simple Question, but I'm finding a lot of dispute on the "lesser" internet. Basically, given a line, is it parallel with itself?
1
vote
2answers
109 views

Definition of identity in a monoid

I'm having trouble understanding the way the identity element is defined in Lang's Algebra. Below is the relevant information. Suppose we have a monoid G with elements $x_{1},...,x_{n}$. We can define ...
4
votes
2answers
284 views

equivalent definitions of orientation

I know two definitions of an orientation of a smooth n-manifold $M$: 1) A continuous pointwise orientation for $M$. 2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$. ...
8
votes
4answers
589 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
2
votes
1answer
114 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
241 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 ...
1
vote
2answers
152 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 ...
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7answers
8k 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
votes
1answer
307 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
512 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 ...
53
votes
8answers
7k 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
320 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
votes
1answer
102 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
votes
4answers
460 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!
1
vote
2answers
416 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
849 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
626 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
211 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
228 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
565 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 ...
5
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
votes
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
votes
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
1k 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 ...
9
votes
3answers
557 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
387 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
118 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 ...
10
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2answers
521 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 ...
3
votes
1answer
282 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
79 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
267 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 ...
4
votes
5answers
2k 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
512 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 ...