Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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7
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1answer
477 views

Terminology for infinite groups, all of whose subgroup have finite index.

Is there a name for (infinite) groups such that every non-trivial, proper subgroup has finite index (e.g. $\mathbb{Z}$)?
7
votes
3answers
254 views

The usage of the term “family” in mathematics

In our lecture notes, the term "family" is used quite persistently and with no definition given. Some examples: (i) Let V be a vectorspace and $(v_i)_{i \in I}$ a family of vectors... ...
7
votes
2answers
2k views

opposite of disjoint

Sets whose intersection is the empty set are called disjoint. What is the opposite of a disjoint set? For example the sets $\{1,2\}$ and $\{2,3\}$ satisfy this condition. I know that you can just say ...
6
votes
1answer
387 views

Difference between elementary submodel and elementary substructure

This is a really "elementary" question, forgive the pun. What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)? Sincere thanks for help.
6
votes
2answers
183 views

History of the vocabulary for group extensions

In regular everyday English if you say something like "A was extended by B to get C", to me it means that A was in existence, B was added onto it, and now there is a larger object C. For example, ...
5
votes
1answer
476 views

Rel: the category of relations

$\text{Rel}$ is the standard name for the category of sets and relations. Confusingly in "Abstract and concrete categories" (ACC), page 22, $\text{Rel}$ is defined as a category whose objects are ...
5
votes
1answer
89 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
5
votes
3answers
428 views

A 1-1 function is called injective. What is an n-1 function called?

A 1-1 function is called injective. What is an n-1 function called ? I'm thinking about homomorphisms. So perhaps homojective ? Onto is surjective. 1-1 and onto is bijective. What about n-1 and ...
4
votes
0answers
110 views

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $M ...
3
votes
1answer
203 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
3
votes
2answers
251 views

What should we call the 'sets' which don't exist under certain set theory axioms?

For example we know that the set of all ordinals does not exist in ZFC, so what should we call it? Set? Collection?
3
votes
2answers
2k views

What is the difference between the terms “classical solutions” and “smooth solutions” in the PDE theory?

What is the difference between the terms "classical solutions" and "smooth solutions" in the PDE theory? Especially,the difference for the evolution equations? If a solution is in ...
3
votes
2answers
1k views

name for a rational number between zero and one?

I'm searching for a unified name to convey for the concept that a number will always be between zero and one. Some info for context: in probability we've got a number between 0 and 1. Percentages ...
2
votes
2answers
1k views

What does a condition being sufficient as well as necessary indicates?

I have a question in a book I am solving(Discrete Structures by Kolman, Busby & Ross). I am unable to make sense from the question. It is stated below, Show that k is odd is a necessary and ...
1
vote
1answer
63 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
1
vote
2answers
2k views

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that "when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as ...
0
votes
1answer
59 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
14
votes
3answers
474 views

Etymology of “topological sorting”

This may be a dumb question, but what's "topological" about topological sorting in graph theory? I thought topology was related to geometry and deformations.
11
votes
3answers
6k views

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it's more a synonym of deducing. Am I ...
11
votes
2answers
6k views

Difference between root, zero and solution.

Can somebody precisely tell me what is the difference between a root, a zero and solution ? Is it correct to say that an equation has solutions, and a polynomial has zeros or roots?
11
votes
3answers
801 views

What is the $x$ in $\log_b x$ called?

In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is ...
10
votes
2answers
705 views

Injection and surjection - origin of words

Can anyone give me a good explanation of how and why words surjection and injection came into use in mathematical community? What do they exactly mean? Who introduced them? I have a feeling students ...
10
votes
2answers
693 views

Usage of the word “formal(ly)”

This is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems to ...
8
votes
1answer
452 views

Why are modules called modules?

I know that a module is a generalization of a vector space, but I would like to know why are modules called modules? Thanks for your kindly help.
8
votes
2answers
400 views

What is an element of a rng called which is not the product of any elements?

Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in ...
7
votes
0answers
162 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
7
votes
3answers
393 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
7
votes
3answers
1k views

What does a space mean?

In Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?
7
votes
2answers
881 views

Meaning of “a mapping factors over another”?

I was wondering what "a mapping factors over another mapping" generally means? Does it have something to do with commutative diagram in category theory? I have seen this usage in different ...
7
votes
2answers
541 views

(k+1)th, (k+1)st, k-th+1, or k+1?

(Inspired by a question already at english.SE) This is more of a terminological question than a purely mathematical one, but can possibly be justified mathematically or simply by just what common ...
6
votes
1answer
383 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
6
votes
1answer
2k views

Function theory: codomain and image, difference between them

Can't figure out the difference between them. I have read wiki article about codomains and images, but what is the difference? It seems confusing the examples part in codomain article. How can we ...
6
votes
5answers
983 views

How common is the use of the term “primitive” to mean “antiderivative”?

I don't know if this should actually be asked on the English stackexchange. It seemed like I would find better answers here. I have all but finished an undergraduate degree in mathematics in the ...
5
votes
1answer
237 views

What is the general term for concepts like length, area and volume?

In geometry, we have concepts such as length (of a 1-dimensional line), area (of a 2-dimensional square) and volume (of a 3-dimensional cube). What is the general term for these concepts, such that ...
5
votes
1answer
140 views

Why do we traditionally use letter U for open sets?

Most of traditional usages of symbols in mathematics have origin in English, German or French words that start with that letter, for an example: $p$ for a prime number, $\mathbb{Z}$ for integers (ger. ...
5
votes
4answers
7k views

What is the difference between an axiom and a postulate?

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more ...
5
votes
1answer
511 views

Difference between elementary logic and formal logic

In Kelley book on topology, in the appendix on elementary set theory, he says in the second paragraph, that "a working knowledge of elementary logic is assumed, but acquaintance with formal logic is ...
5
votes
1answer
252 views

Alternative name for “closed set”

It is usually argued (and also joked about) that classifying sets into open and closed is a bit paradoxical, since sets can be open and closed at the same time, or neither. This can be analyzed very ...
5
votes
3answers
4k views

What is Modern Mathematics? Is this an exact concept with a clear meaning? [closed]

Motivated by this question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern ...
4
votes
1answer
111 views

On group theory terminology

Let $G$ be a finite group. Consider the next number $$m(G):=\min\{m\in\mathbb{N}\mid G\ \text{can be embedded into}\ S_{m}\}.$$ It is obvious that Cayley's theorem yields $m(G)\leq |G|$. My ...
4
votes
2answers
353 views

Is there a meaningful distinction between “inclusion” and “monomorphism”?

The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case? Edit: ...
3
votes
1answer
79 views

Why are free modules called “free”?

Let $R$ be a ring (not necessarily commutative) with multiplicative identity. A $R$-module $M$ is called free if $M$ has a linearly independent generating set $\beta\subseteq M$. That is, for any ...
3
votes
1answer
108 views

Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?

Fix a positive integer $p$, possibly prime. For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow ...
3
votes
0answers
44 views

What's a concise word for “the expression inside a limit”? Limitand?

In $\sqrt {f}$, $f$ is the radicand. In $\sum g_i$, $g_2$ is a summand. In $x \times y \times z$, $y$ is a multiplicand. In: $$\displaystyle \lim_{n \to +\infty} h_n(x)$$ or: $$h(x) \to \ell \quad ...
3
votes
1answer
103 views

What do you call a set whose subsets all have unique sums?

An example would be $\{1, 3, 7\}$, which has subsets with sums $1, 3, 7, 4, 10, 8, 11$. What is this called?
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votes
0answers
112 views
+150

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
3
votes
0answers
196 views

Arithmetic and geometric sequences: where does their name come from?

Where does the name of these two famous types of sequences come from? The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the ...
3
votes
0answers
183 views

term for a sum of diagonal and skew-symmetric matrix?

Is there a term for a matrix that is a sum of a diagonal and a skew-symmetric matrix? One particular example of this is a 2x2 matrix of the form $$ M = \begin{bmatrix} a & b \\ -b & a ...
3
votes
3answers
947 views

Path components or connected components?

Can anyone explain the difference between these two terms? Are they basically different names for the same thing or totally different things?
3
votes
1answer
85 views

Minimality in the case of partial derivatives and Sobolev spaces?

I am trying to understand this question here that considers Sobolev spaces apparently and hence partial derivatives. What is the definition of minimality there? Is the minimality defined by ...