2
votes
1answer
32 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
0
votes
1answer
45 views

Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
3
votes
1answer
64 views

What is a 'disjunct' of a union called?

Say I have a set $C = A \cup B$ and I want to refer to $A$ in natural language. Had the expression been a Boolean formula with a disjunction, then I would call $A$ the first disjunct. Is there a ...
1
vote
1answer
41 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
1
vote
1answer
23 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
1
vote
0answers
19 views

Is there a name for the corresponding notion of inductive subset in the context of well-ordered sets?

This is a question of terminology. I can't avoid being a little verbose before getting to it. The principle of mathematical induction states that, for any subset $e$ of $\omega$ (the set of natural ...
5
votes
1answer
338 views

Term for: There Exists a Rational between every two Rationals?

The integers and the rationals have the same cardinality, but the rationals satisfy the property that: $$ \forall p,q\in\mathbb{Q},\quad \exists r\in\mathbb{Q}\quad \textrm{s.t.}\quad p<r<q, $$ ...
4
votes
2answers
80 views

Mathematical concept for formal languages

A formal language is defined as a subset of finite-length strings over an alphabet. It is similar to the mathematical concept "relation", but the lengths of its strings are not fixed. Since the name ...
0
votes
0answers
15 views

How to write the condition for Image of a function?

If $\Omega_l$ is $\Omega$ with $|x|<l$ and if $\Omega_S$ is the image of $z$ under mapping how we will write the condition for it. Am I right if I write $\Omega_S$ is $\Omega$ with $|S|<l$ or ...
0
votes
1answer
36 views

$\mathcal N (A):=\mathcal P(A)-\varnothing$ notation

Define $\mathcal N$ $\mathcal N (A):=\mathcal P(A)\setminus\{\varnothing\}$ Does $\mathcal N$ has a special name and standard notation?
0
votes
0answers
58 views

The symbol $\mathcal P(\alpha)$ where $\alpha$ is a cardinal

$X$ is a set. There's a term: ‌$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ the power set $\mathcal P(2^{|X|})$ in an article. What does it mean? It seems $2^{|X|}$ is the power ...
0
votes
1answer
11 views

An indexed family of filters and their elements

Let $X$ is an indexed (by some set $n$) family of filters (on some poset $\mathfrak{A}$). Is there any standard notation/terminology for the set $\{ y\in \mathfrak{A}^n \,|\, \forall i\in n:y_i\in ...
1
vote
0answers
30 views

Name of the “left” set on which a partial function $f\colon \mathbb N \times \mathbb N \to\mathbb N$ is defined

Given a partial function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ Does the set: $$A = \{ x \in \mathbb{N} \mid \exists y \in \mathbb{N} \text{ such that } f(x,y) \text { is ...
0
votes
4answers
88 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 ...
3
votes
2answers
63 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, ...
2
votes
5answers
99 views

Question about definition of binary relation

Wikipedia says: Set Theory begins with a fundamental binary relation between and object $o$ and a set $A$. If $o$ is a member of $A$, write $o \in A $. I thought that a binary relation is a ...
2
votes
1answer
77 views

What if union of disjoint sets results in universal set?

I have a question related to set theory. If $A_1,A_2,A_3\dots, A_n$ belongs to universal set $U$, and if all of the sets are disjoint i.e. $A_i \cap A_j = \emptyset$ for all $i$ and $j$. And If ...
1
vote
2answers
99 views

Initial Segments and Initial Sections of Posets

For a set A with a partially ordering <=, define the following 1) A subset s(x) of A = {y in A such that y <=x} 2) A subset S of A with the property that for every x in S then all y in A ...
2
votes
2answers
63 views

Correspondence as a graph of a multifunction

Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do ...
0
votes
0answers
39 views

Identity relation of many variables

The identity relation on a set $A$ is $\operatorname{id}_A = \{(x;x) \,|\, x\in A\}$. This can be generalized for any (possibly infinite) index set $N$ as $\{(\lambda i\in N: x) \,|\, x\in A\}$ (here ...
0
votes
1answer
25 views

Partial order up to equivalence

In certain contexts one runs into something like a partial order, but the antisymmetry property is weakened as follows: if $x \preceq y$ and $y \preceq x$ then $x \simeq y$, where $\simeq$ is a given ...
2
votes
1answer
668 views

into function vs injective function

In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...
1
vote
2answers
58 views

Basic Cartesian prodcuts

I am having some issues grasping basic ideas of Cartesian products. I am reading a PDF my professor gave us explain sets/Cartesian products. If $\mathbb{R}\times \mathbb{R}$ can be written as ...
2
votes
1answer
87 views

The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
1
vote
2answers
122 views

What is the name of this set?

What is the standard name for the set of all n-ary functions, where n is a natural number,of some set S, say the reals or the complexes? We have the notation S^S, but that is only the set of 1-ary ...
-1
votes
2answers
33 views

What is a set of overlapping sets?

If I have a set $X$ and a set $Y$ and $\forall y \in Y : y \subseteq X \land \exists y_1, y_2 \in Y : y_1 \ne y_2 \land y_1 \cap y_2 \ne \{\}$, what is the relationship between $X$ and $Y$ called?
0
votes
4answers
49 views

Name of a set of the form {x,y}

I know that a doubleton is a set with exactly two elements, but what is the name of a set with either exactly 1 element or exactly 2 elements? In other words, what is the name of a set of the form ...
3
votes
1answer
90 views

Given a subcollection of a powerset, do these “separation” relations have names?

Let $X$ denote a set and $\mathcal{F}$ denote a subcollection of $\mathcal{P}(X).$ Do the following relations on $\mathcal{P}(X)$ have a name? For $A,B \subseteq X$, call $A$ partially separated from ...
2
votes
1answer
65 views

Is this “set quotient” known?

Let $A,B$ be subsets of a set $X$. Then there is a largest subset $C \subseteq X$ such that $C \cap A \subseteq B$. Explicitly, we have $C = \{x \in X : x \in A \Rightarrow x \in B\} = (X \setminus A) ...
0
votes
0answers
41 views

Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A ...
-2
votes
1answer
55 views

Problem in Set Theory determining elements [closed]

"H does not include D" in this statement is D is an element or a set? if it is a set what is the set notation?
1
vote
5answers
232 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
2
votes
1answer
48 views

“Set” vs “collection” terminology: what is the difference?

Can someone tell me what is the difference in saying $A$ is a set of even numbers and $X$ is a collection of even numbers ?
7
votes
0answers
87 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
1
vote
2answers
97 views

“Collection”: What does it mean?

I've seen a lot of question of same ilk as the request I'm about to pose, but what I'd like to know is what does "any collection" mean in the following request: Prove that the intersection of any ...
5
votes
0answers
98 views

The counted is to the countable as the ??? is to the (order)-isomorphic.

We sometimes need to distinguish the counted from the countable. A counted set is a set equipped with a particular bijection into (some of) the natural numbers; a set is countable if there exists such ...
0
votes
1answer
80 views

Name for Cartesian Product variant that does not return an empty set if one of the sets is empty

I am looking for the name of this mathematical operation that behaves very similar to Cartesian Product. Given: A = {1,2} ...
2
votes
3answers
62 views

Name for $X^\infty=\bigcup\limits_{k=0}^\infty X^k$

I'm making structures associated with groups, rings and so on in OCaml and in order to do so I started by defining sets and a few operations (intersection, union, difference, carthesian product, ...
2
votes
2answers
103 views

The union of a countable set of countable sets?

Let $A$ be an countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\ldots,a_n\right)$ $B_n$ is the union of a countable set of countable sets. This question maybe about the ...
0
votes
1answer
48 views

Formal notation when using the axiom of specification

The axiom of specification states formally that for every property $\varphi$ holds $\forall X\exists Y\forall x(x\in Y\longleftrightarrow x\in X\wedge\varphi(x))$. Since from the axiom of ...
2
votes
0answers
49 views

Do these union- and intersection-like operations have a name?

I have two sets of pairs, e.g: $$A = \{ (a, 1), (b, 2), (c, 3) \}$$ $$B = \{ (b, 12), (c, 13), (d, 14) \}$$ I also have two operators which match the first element of pairs and return a pair of ...
1
vote
2answers
24 views

Suppose A is a set of (x, y)', what is the name of the set that consists of all x in A?

Let $A$ be a set of a vector $(\mathbf{x}',\,\mathbf{y}')$. Here $\mathbf{x}'$ and $\mathbf{y}'$ could both be vectors. Is there a particular terminology for the set of all $\mathbf{x}'$ in the set ...
19
votes
4answers
399 views

“$f$ is a function from $A$ to $B$” vs. “$f $is a function from $A$ into $B$”?

When we say that $f$ is a function from $A$ to $B$ is this different from saying $f$ is a function from $A$ into $B$ I know what injective ("1-1"), surjective ("onto"), and bijective ...
1
vote
5answers
775 views

Define Onto and one to one meaning

i understand what one to one means. However, im struggling with understanding of onto Can anyone give me an example of onto? i want to understand what onto means.Can anyone explain what onto means in ...
2
votes
1answer
65 views

Is there a name for this type of relation?

Let $S$ be a set. Let $\sim$ be a binary relation on $S$. Suppose $\sim$ follows these three rules. $x\sim x$ for all $x\in S$ (reflexivity). If $x\sim y$, then $y\sim x$ for all $x, y \in S$ ...
5
votes
2answers
110 views

Is the cardinality of a set necessarily a natural number?

I've never seen phrases like "$\sqrt{5}$ people" or "a set with $\pi$ many elements". Are there sets with cardinality, say, $\frac{1}{2}$? Edit: As Brian M. Scott pointed out, the only real numbers ...
1
vote
5answers
700 views

How do I find the image of the functions $y=2$ and $y = 2x - 6$?

The function is $y=2$, the domain is just 2? And the image of it? I don't think I quiet understand what the image of a function means, the domain is all values that it can assume, correct? Could you ...
3
votes
3answers
88 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 $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
2
votes
1answer
96 views

Term for sets with a bijection between them

If there exists an isomorphism between $G$ and $H$, we say that $G$ and $H$ are isomorphic. If there exists a bijection between $A$ and $B$, we say that $A$ and $B$ are _______. Is there a ...
2
votes
2answers
95 views

Real definition of “countable set”

Is there any correct definition for countable set? I read some book saying a set is countable if there is a bijection between it and the set of all natural numbers, while some other text says if there ...