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

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4
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5answers
567 views

How can I succinctly but correctly say that a set is finite?

If I want to say that a set $A$ is numerable but infinite, I can do so like this: $$|A| = \aleph_0$$ What should I use instead to say that a set is finite? $|A|\in\mathbb{N}$? $|A|< \infty$? ...
3
votes
2answers
421 views

Validity vs. Tautology and soundness

I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula? They also ...
3
votes
3answers
100 views

What are the sets $S_n=\omega-n$ called?

What are the sets $S_n$ where $S_n:=\omega-n$ called? I explain better: if ordinals are defined in this way $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
3
votes
3answers
117 views

What does “lower density” mean in this problem?

If $\mathscr{U}$ is a ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. This is an exercise on page 76 of Problems and Theorems in Classical Set Theory, ...
3
votes
2answers
241 views

Zorn's Lemma $\equiv$ Axiom of Choice

I'm confused a little bit about this, I've been told many times that Zorn's lemma is equivalent to the axiom of choice. Is it an axiom or is it lemma, I mean is there a proof of Zorn's lemma or we ...
3
votes
2answers
2k views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
3
votes
1answer
77 views

Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.

I will illustrate my question in the case of the definition of vector spaces. It is custom to define a vector space in the following way: "Let $K$ be a field. Then a $K$-vector space is a set $V$ ...
3
votes
2answers
2k views

Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
3
votes
2answers
185 views

Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is?

Is a 2-dimensional subspace in a 7-dimensional space still called a plane? I know that a 6-dimensional space in 7-dimensional space is called a hyperplane because the difference in the number of ...
3
votes
1answer
1k views

What does “relatively closed” mean?

Let $S\subset U$. What does it mean to say that $S$ is relatively closed in $U$? Also $U\subset\mathbb{R}^{n}$ is open and bounded, but I don't know if that's essential. Here follows an example from ...
3
votes
2answers
281 views

What is the name of the function $f(k) = \text{max}(k,0)$ on $\mathbb Z$? [duplicate]

Let $k\in \mathbb Z$; is there a name for the function $f(k)$ below? $$ f(k) = \text{max}(k, 0) $$
3
votes
2answers
363 views

Square for $x^2$, Cube for $x^3$, Quartic for $x^4$, and what's for $x^1$?

What's the general form for $x^y$? What's the specialized form for $x^1$ and $x^0$?
2
votes
1answer
48 views

What is the name of this terminology?

Let $G$ be the group generated by a set $X=\{x_1,\cdots,x_n\}$. Then each element can be (not necessarily uniquely) written as a product of the form $x_{j_1}^{e_1}\cdots x_{j_k}^{e_k}$, where each ...
2
votes
3answers
261 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
2
votes
3answers
92 views

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in ...
2
votes
1answer
75 views

Name for numbers expressible as radicals

What is the name (there must be one!) for real (or perhaps complex) numbers expressible as radicals? Radical numbers? Solvable numbers? (following the same logic as ‘solvable group’). In other ...
2
votes
0answers
564 views

Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
2
votes
1answer
56 views

Term or factor?

I was reading a book on probability and encountered a summation expression like $$P(Y\mid X, Z) = \sum_{W}P(Y\mid X, Z, W)P(W\mid X,Z)$$ followed by the author referring to "terms of the summation". ...
2
votes
2answers
633 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
2
votes
1answer
105 views

Definition of “totient”

I had always taken the term "totient" to be defined by saying that the totient of a positive integer $n$ is the number of positive integers less than $n$ that are coprime to $n$. Thus, for example, ...
2
votes
2answers
132 views

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates ...
2
votes
2answers
140 views

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
2
votes
1answer
44 views

What is the names of $A\vec{x}=\vec{b}$ linear equation system components?

Having $A\vec{x}=\vec{b}$ . What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
2
votes
1answer
200 views

What is the purpose of defining the notion of inflection point?

What is the purpose of defining inflection point? I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes. It has to have some purpose ...
2
votes
2answers
123 views

Terminology: $H$ and $K$ are subgroups. What is $HK$ called?

Let $H, K\leq G$. I was wondering what you call the "product" $HK$ of $H$ and $K$. I was trying to verbalise the steps of showing $G$ is a semidirect product: Normality of $H$: $H\unlhd G$. Trivial ...
2
votes
1answer
89 views

Name of corresponding objects in equivalent categories

This question is only about terminology. Inside a category we have the standard wordings: An arrow $f: X \rightarrow Y$ is an isomorphism if there is another arrow $g: Y \rightarrow X$ such that $g ...
2
votes
4answers
544 views

Does the word “integer” only make sense in base 10?

Does the word "integer" only make sense in base 10? I've always wondered this and have never seen it really discussed anywhere. We all understand the typical definition of an irrational number, ...
2
votes
1answer
117 views

Terminology for a property that holds in the finite but not infinite case?

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don't ...
2
votes
1answer
1k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
1
vote
1answer
55 views

Is there a name for this result in planar geometry?

I found out that the following statement is fairly easy to prove: Let $A$, $B$ and $C$ be thee distinct points in the plane. Let $S_{AB}$ be the circle that has the line segment $AB$ as a ...
1
vote
1answer
119 views

Terminology: Delta vs… absolute?

Delta is the change in a value. Using the term "delta" on the one hand, how, on the other hand, would you refer to the base value from which the given delta is derived? Is there a more precise term ...
1
vote
2answers
65 views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
1
vote
1answer
28 views

Need help in understanding $ord_p{a}$ as used in Theorem 1.1 from “On Some Exponential Equations Of S. S. Pillai”

I have a question about very early argument in the proof of Thereom 1.1. Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then ...
1
vote
1answer
52 views

Correct Terminology in the Context of Rings

Suppose I have a ring $(A, ◦, •)$ where $A$ is a set of elements $\{α, β, γ,\ldots\}$. Can $◦$ and $•$ with which the ring is equipped be properly termed, in English, its "internal laws of ...
1
vote
0answers
42 views

A map from one set of words to another that is not a morphism

A word is a concatenation of letters from a non-empty set called an alphabet. For example, if the alphabet is $\{a,b\}$, then $bba$ is a word from that alphabet. Let the set of all finite words made ...
1
vote
2answers
62 views

how do you call a function that breaks down on y?

How do you call a (linear) function (or the point), which breaks down to 0 on ordinate (axis y), as soon as you breach a certain x1 value?
1
vote
1answer
56 views

Recursive application of a function : Symbol of [duplicate]

I need to apply a function $f(x)$ recursively/repeatedly for n times; how do I express it (mathematically) ? Is their a mathematical symbol which denotes $f(x)$ applied n times ie $g(x,n)$ ...
1
vote
1answer
99 views

Difference: Proposition and Observation

As far as I know, a proposition is a statement which might be used to prove a theorem but is also of independent interest. How would you differentiate it from an observation? Would you say it is a ...
1
vote
1answer
143 views

Complement and Negation: $P(A)=0\rightarrow P(\neg A)=1$?

My earlier question became too long so succintly: Suppose $P(C)=0.2$. Its complement is 0.8 i.e. $P(C)^C=0.8$ but what does $P(¬C)$ mean? I think I am messing up the term complement and negation? ...
1
vote
3answers
973 views

Is there a special name for the operands of a multiplication?

Sometimes operands for a specific operation are given a special name. For example, in division the first operand is a quotient, the second is a divisor. Is there a word that means "one of the operands ...
1
vote
1answer
132 views

Sufficient/necessary vs. weaker stronger

I know what sufficient resp. necessary means, but I'm confused, when our professor uses the terminology weaker resp. stronger. I couldn't yet find out, what the translation of these pair of words to ...
1
vote
2answers
67 views

Is there a particular notation for a function confined in a set?

For example $f:\mathbb{R}\to\mathbb{R}$ is a function. How to simply express a correspondent function $g:\mathbb{Q}\to\mathbb{R}$ such that $g(x)=f(x)$, $\forall x\in\mathbb{Q}$?
1
vote
1answer
118 views

Foam-like graphs

What's the "official" name of a connected planar graph consisting entirely of polygons (cycles), glued together at edges, e.g. - among other things - without "end vertices" (of degree 1) and without ...
1
vote
1answer
74 views

Notation and naming for two operations with $p$-form valued $n$-forms

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much. I'm ...
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1answer
91 views

Whether to use 'OR' or 'AND'

My doubt is: while solving equations or inequalities consisting of absolute values when should we use the conjunction 'OR' and when to use 'AND'? whats the difference between them ?
1
vote
1answer
270 views

Is there a different name for strongly Darboux functions?

A function $f\colon\mathbb R\to\mathbb R$ is called Darboux function, function with Darboux property or function with intermediate value property for for any $a<b$ and any $z$ between $f(a)$ and ...
1
vote
1answer
114 views

Bernoulli Distribution with support different from $\{0,1\}$

Suppose the support of a distribution is $\{12 , 13 \}$ with $P(X = 12) = p$ and $P(X = 13) = 1-p$. Is this still a Bernoulli distribution even if the support is not $\{1, 0 \}$?
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vote
1answer
171 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
1
vote
1answer
268 views

What does Linear Congruential mean?

How does one interpret the terms "Linear" and "Congruential" as in a "Linear congruential RNG"? I am used to linearity by $f(ax)=af(x)$. This does not seem to me to hold true in this case ($\bmod$). ...
1
vote
2answers
946 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 ...