0
votes
2answers
73 views

$\Bbb Z^\ast$ What is this notation?

What does $\Bbb Z^\ast$ mean? I would think some subset of the integers but I cannot find a definition. Thank you.
0
votes
0answers
21 views

Set-builder defining set size and valid elements? Notation help.

I'm working on some documentation for a system that uses an arbitrary waveform generator that accepts data with a number of requirements. I would like to include some formal definitions of the ...
0
votes
2answers
23 views

Cannot understand a mapping function which include some sets

I have kept a screenshot of my problem below which describes about the various sets. At the last line, there is an expression where a function delta uses those sets and maps them into another. I am ...
2
votes
2answers
54 views

Element of a Singleton (set with one element) notation

I was wondering what the notations are for indicating the element of a singleton (or unit set, or set with cardinality 1). This would be the inverse of set construction: $$X = \{y\} \tag{1}$$ $$y = ...
1
vote
1answer
49 views

correct Set theory notations.

What is correct notation for the following, I have seen both in some books. To show an empty set, is it Φ(phi) or Ø(slash O) or both. To show an Universal set, is it ε(epsilon) or U or both. I am ...
3
votes
1answer
49 views

$A \setminus B \cup C = A \setminus (B \cup C)$? [duplicate]

$A \setminus B \cup C$ or $A \setminus (B \cup C)$? Sorry as this is a very soft question, but I couldn't find the answer anywhere. Are these two things generally considered the same?
1
vote
1answer
23 views

Small notation question about union of chains (Set Theory)

The question is derived from this question I encountered: Let $A$ be a set, and let there be a function $f: A \rightarrow A$, so that for every $a \in A$, $f(a) \neq a$. Define $S=\{X \subseteq A: ...
0
votes
1answer
30 views

Reading set notation for the flow number

Could someone please help me to understand the following notation: Flow Number: In case of need, S refers to a set of flows, C(s) is called congestion and D(S) is dilation. How would you put in ...
0
votes
1answer
19 views

Notation for a collection of sets under a certain condition

I am looking for the notation to describe "A collection of sets that are the union of a finite number of intervals". Is this correct - $A = \{A_i\}_{i \in I}$ where each $A_i = \bigcup_{n \in N} ...
0
votes
2answers
21 views

How do you write the set where $2$ numbers are chosen

A = the event that the sum of outcomes of $2$ dice being thrown So do I say $A = \{14;41;23;32\}$ That looks like I'm saying $41$ (the number) not $4$ from one die and $1$ from the other which ...
0
votes
0answers
29 views

Notations for elementary set theory

I wish to understand the following notations which defined for every cardinals, $a,b$: $P(a,b) = \left| In(A,B) \right|$ $C(a,b) = \left| P_b(A) \right|$ $E(ab) = \left| Eq(A,B) \right|$ ...
1
vote
2answers
79 views

What does $\vee$ mean in set theory?

The following proof is from Probability by Davar Khoshnevisan. There is a symbol $\vee$ in the third sentence of the proof. What does this symbol mean, please? There seems no definition about it in ...
2
votes
3answers
90 views

Notation for a space of finite sequences

For a given set $X$, what is the notation for the space of all finite $X$-valued sequences? I realise that the space of $n$-tuples is written as $X^n$, and the space of infinite sequences is ...
1
vote
0answers
18 views

Is there notation or a name for the complement of the unbounded face of a planar graph?

Let $G$ be a finite graph embedded in $\mathbb{C}$. Let $F$ denote denote its unbounded face. Is there notation or a name for $F^c$ without referring directly to $F$. Of course this is equivalent ...
2
votes
1answer
20 views

Domain of a composite function

I was given the question: Find the domain of the function $f(x)=\ln(\ln(\ln x))$ I found the answer by inspection: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
1
vote
0answers
21 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
0
votes
1answer
24 views

Understanding the definition of the d-dimensional Hyperube

Please see the picture bellow about the definition of the nodes of the d-dimensional Hypercube. Could anyone please tell me what does that notation means. I get confused with the superscript after the ...
3
votes
1answer
36 views

Notation for a set of objects expressed by symbols

I have two questions about set notation The first one is how you would write down a set of objects expressed by symbols? Let's say we have $n$ persons identified by $p_i$. Can I write: $$P = \{p_1, ...
3
votes
3answers
75 views

Notation for choosing the k smallest elements from a set of integer

Is there any specific notation for picking $k$ elements from a set which are the smallest? Ex: {$1,3,5,7,9,11$} with $k = 3 \Rightarrow$ We want $1,3,5$
0
votes
3answers
65 views

Set question - $ ℤ^+ = ℕ$ [duplicate]

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$ if not, why? Thank you in advance! I appreciate your help!
1
vote
1answer
28 views

Set Theory Elementhood Notation

From How to Prove it: Given $A=\{n^2|n \in N\}$ where $N$ is the set of all natural numbers. I want to express A in terms of elementhood test notation. Velleman says $A=\{x| \exists n \in N ...
3
votes
2answers
58 views

$\left(A\times B\right)^n=A^n\times B^n$?

Is this property true: For any set $A$ and $B$: $$\left(A\times B\right)^n=A^n\times B^n?$$ Where $A\times B$ is the Cartesian product and $A^n=\underbrace{A\times A\times \cdots\times A}_{n\, ...
0
votes
1answer
48 views

How to write $A\cup\emptyset\times A\cup\emptyset$

How to write the cartesian product of this? $$(A\cup\emptyset)\times (A\cup\emptyset)$$ Is it: $$A^2\cup\emptyset^2?$$ What does $\emptyset^n$ means?
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
1answer
43 views

Notation question in elementary set theory

Let $A$ be a set. What is defined as $UA$? Is it the union of all sets that $A$ includes? Could someone provide an example for this notation? Thank you!
2
votes
1answer
44 views

Notation in set theory.

What is the difference between these two notations? $\{a_k\}_{k\in K}$ $\{a_k, k\in K\}$ Is this correct? $\{a_1\}$, $\{a_2\}$, $\dotsc$ $\{a_1, a_2, \dotsc\}$ Or 1. and 2. are the same? Is ...
0
votes
3answers
30 views

How can I express, “The set of integers greater than x and less than y”?

I know I could express it this way (x = 0, y = 10): $$ \{ 1, 2, 3, 4, 5 , 6, 7, 8, 9 \} $$ in simple cases. This is what I could come up with for the more general case: $$ \mathbb I = \{ i_n | i_n ...
0
votes
4answers
67 views

“$((A\times B) \to C)$” denotes what?

I'm having some trouble understanding notation. The question is For any three sets $A,B,C$ , $((A\times B) \to C) =_c (A \to (B\to C))$ Exactly what does "$((A\times B) \to C)$" denote? Is ...
1
vote
3answers
44 views

What does $f: 2^{\mathcal{S}}\rightarrow\,\mathbb{R}$ mean?

A function $f: \mathcal{S}^n\rightarrow\,\mathbb{R}$ This is I understand. $x\in\mathrm{dom}\,f$ means that $x$ is a vector of size $n$ where its elements are taken from the set $\mathcal{S}$. ...
7
votes
1answer
105 views

First usage of the symbol ∈

Concerning a book [1] I am reading the symbol $\in$ was first used by Giuseppe Peano and is the first letter $\epsilon$ (epsilon) of the word ἐστί (means "is"). Does anyone know in which work of Peano ...
0
votes
1answer
36 views

Set notation of $S^1 \times S^1$

This is a simple question, but should this be written as: $\hspace{120pt}S^1 \times S^1 = \{(z_1,z_2)\in\mathbb{C}\times\mathbb{C}:|z_1|=|z_2|=1\}$
0
votes
2answers
37 views

How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$ ( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ ) $$
2
votes
2answers
96 views

What does $\in$ mean?

I'm reading a textbook on complex analysis and I've come across notation using this ($\in$) symbol. In the context of "an argument of $z = x + iy$ is a number $\phi \in \mathbb R$ such that $x = ...
3
votes
1answer
110 views

What is $\mathbb{R}^\mathbb{R}$

I do not know what it is. $\mathbb{R}$ is the set of real numbers. How come $\mathbb{R}\times\mathbb{R}\times \ldots $? Thanks.
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
votes
1answer
39 views

What is the place holder glyph for a set?

What glyph do set theorists use to denote an unspecified set? For example, logicians use φ to talk about an unspecified sentence in first order logic. Does set theory have a comparable glyph? Thank ...
-1
votes
3answers
84 views

Set theory symbol

I'm studying very basic set theory for a module and have come across this symbol: | quite a few times, although I have no idea what it means, can someone explain what it is and how it makes sense in ...
1
vote
2answers
147 views

discrete math: is there a difference between $\subseteq$ to $\supseteq$

I have a question which asks if $X \supseteq I$. Is it just the same as $I \subseteq X$? Because I never saw it the other way around or learned about it, I'm confused.
1
vote
1answer
47 views

Notation for “set of all possible unions”

For a set $S$, for "all possible subsets of $S$" you have $\mathcal{P}(S)$. For a set $S$ consisting of sets, for "the union of all sets $T\in S$" you have $\bigcup_{T\in S}T$. Is there a notation ...
0
votes
1answer
20 views

How to mention a specific member of a set?

I know there are other questions like this, but they are yet to satisfy me. How would would go about mentioning specific members of a set. For instance, if I wanted to mention the third element from ...
1
vote
2answers
183 views

A Theorem About Compactness and

My first exposure to any sort of topology is from Spivak's Calculus on Manifolds. I think I understand compactness conceptually, I'm just finding the rigor a little bit elusive. My first question ...
0
votes
1answer
21 views

Formal way to add a set to an existing collection

If we have an existing collection of sets $\{G_\alpha\}$ (possibly uncountable) and a $X$ that we would like to add to the collection, what is the formal way to do this? Note that I'm not looking for ...
1
vote
2answers
42 views

Could someone explain this notation?(set theory)

The expression is : $$\bigcap_{s \in S}L_s$$ Where $s$ means student, $S$ is the set of all students and $L(x, y)$ means "$x$ likes $y$". So where I'm reading it from ("How to Prove It A Structured ...
0
votes
2answers
36 views

Notation question about defining a set

If I'm given: $\{ n^2 + n + 1 \mid n \in \mathbb N\} \subseteq \{ 2n + 1 \mid n \in \mathbb N\}$ Does this mean: "The set $A$ where each element is made by putting a natural number in that formula ...
1
vote
2answers
89 views

Sigma hierarchy of logical formulae

In some papers on mathematical logic I've found references to hierarchy like $\Sigma_1^0$-sentence and so on. What does it mean? What is $\Sigma_n^m$, what is $n$ and $m$ here?
-1
votes
1answer
35 views

What is P(Y) here?

A multivalued map, f: X -> Y, from a set X to a set Y, is a map f: X -> P(Y). Multivalued maps will be also called multimaps. I don't understand what a multimap is in category theory and I think the P ...
2
votes
2answers
74 views

Set builder notation

I'm not sure of the correct notation if someone could please help. Say you wish to have a set comprised of the union of two sets, such as $$h=\Big\{ \begin{bmatrix}x & y\\y & x\end{bmatrix}: ...
1
vote
1answer
35 views

Constructing sets involving predicates. Let $P(x),Q(x)$ be predicates over a set $X$?

Let $X$ be a set and $P(x),Q(x)$ be predicates over $X$. Consider the sets $$Y = \{y\in X\mid P(y)\}$$ $$Z = \{z\in X\mid Q(z)\}$$ Complete the following sentences with quantified propositional ...
2
votes
2answers
50 views

Correct notation to indicate multiplying all elements within a set

Is there a correct notation to indicate multiplying all elements within a set? For example, if $M = \left\{n_0, n_1, ..., n_t\right\}$ be the set of elements where I want to multiply all the numbers ...
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 ...