0
votes
1answer
8 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
29 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
83 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 ...
2
votes
5answers
79 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
64 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
89 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 ...
0
votes
0answers
5 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
30 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
20 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
269 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
49 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
54 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
114 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
29 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
87 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
39 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
53 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
212 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
44 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 ?
6
votes
0answers
57 views

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

If we have a sum $\sum_{i=1}^na_i$, 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 various ...
1
vote
2answers
87 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
89 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
73 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
61 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
98 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
48 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
390 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
442 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
63 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
108 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
584 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
75 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
86 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 ...
1
vote
1answer
46 views

Multiple Indexing Sets

I suspect that this is a trivial thing, but I can't seem to find an example. Say I have some Set indexed by Set $I$: $$\{ x_i \}_{i \in I}$$ This called a "Family" of Sets, correct? Next, say I ...
0
votes
1answer
23 views

Function returning the set of sets which are no superset of each other

Given a set $S$ of sets. Assume a function $f$ which removes all supersets of any set in $S$. Example: $f(\{\{ a\}, \{ a,b\}, \{ a, b,c\}, \{ b,c\},\{ b,c,d\} \}) = \{\{ a\}, \{ b,c\} \}$ Does this ...
6
votes
3answers
140 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... ...
1
vote
1answer
100 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? ...
11
votes
1answer
158 views

What does it mean for a set to have “structure”?

I understand that a set is like a list of things, except that the order doesn't matter and that you can't have any duplicates in a set. For example: $\{3, 1, 4, 2\}$ is the same set as $\{1, 2, 3, ...
4
votes
4answers
364 views

Is it proper to say that two infinite sets are the “same size” if there is a bijection between them?

I get the fact that a set can be called countably infinite if it can be bijected with $\mathbb{N}$, but it feels wrong on many levels to say that they are the same size. Example: $A=\{x \in ...
5
votes
4answers
1k views

What is the difference between a Subgroup and a subset?

What is the difference between a Subgroup and a subset? I know hardly any Abstract algebra, just some things from youtube and wikipedia, but the notion of a subgroup being part of a larger group and a ...
3
votes
1answer
149 views

Difference between “measure on” and “measure over”

I want to make sure I understand the difference between the terms "measure on" and "measure over," assuming there is one. Is a measure on the set $X$ the same as a measure over its power set ...
0
votes
1answer
109 views

Correct reading of Set builder Notation?

could anyone please let me know the correct reading(sentence form) of set builder notation, confused with different interpretation in different resources. Many Thanks
2
votes
2answers
64 views

Is there a name for the set $\{T,F\}$?

Is there a name for the set containing the two Boolean values, i.e. $\{T,F\}$? I am also thinking if $B = \{T,F\}$, and $B^n = \underbrace{B \times B\times B ... \times B}_n$, then is there a proper ...
5
votes
1answer
123 views

Name of a set that allows repetition

If a set cannot contain repetition, what would be the proper term for a group of items that allowed repetition?
4
votes
1answer
93 views

Is there a name for a set none of whose members is a proper subset of another?

Is there a commonly accepted name for a set of sets, $S$, with the property that $s1, s2 \in S$ and $s1 \subseteq s2$ then $s1 = s2$?