1
vote
2answers
35 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
1answer
68 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 = ...
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
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 ...
0
votes
0answers
78 views

What do these symbols mean in set theory?

I don't understand what these symbols mean in this context. I thought the apostrophe was essentially a not, but I don't know why 0 isn't in the answer. I don't know whether T means not. No idea what ...
0
votes
3answers
55 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 ...
0
votes
2answers
137 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.
0
votes
0answers
15 views

argmax over set membership probability

I'm trying to find the argument which maximises a function, where the variables are set membership relations: $\{\in ,\notin \}$. This is how I'm doing it now (a simplified argmax): $$ ...
1
vote
1answer
44 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
16 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
171 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
32 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
58 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
34 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
49 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}: ...
0
votes
1answer
32 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
31 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
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 ...
2
votes
2answers
92 views

What does “\” mean in math

In a Linear Algebra textbook I am reading, the following is stated: $b\notin \operatorname{span}(A \cup \{a\})\setminus \operatorname{span}(A)$. It does so without explaining what "$\setminus$" means. ...
0
votes
1answer
40 views

Notation for a subset of a powerset

If we have a set A containing all the integers between x and y, what is the correct notation for the subset B of the powerset of A where the sets contain between n and m elements? For example if x=1, ...
1
vote
1answer
52 views

Ambiguous set-builder notation

I apologize in advance for the Python code below. I don't know how to express this question unambiguously in the language of math. Let $\mathcal{F}$ be a family of sets. It seems to me that $$S(x) = ...
0
votes
1answer
31 views

Meaning of this notation - $T = \bigcup_{B \in \beta}B$

$$T = \bigcup_{B \in \beta}B$$ Does the $B \in \beta$ mean that $T$ is a union of some arbitrary subsets of $\beta$, or does it mean that $T$ is the union of all subsets of $\beta$?
1
vote
2answers
58 views

What is this form of 'notation' called?

I was reading some of Max Tegmark's lecture materials and I found this little thing. Is there a name for it? Specifically, I am talking about $S_1$ R $S_2$ & $S_1$ R $S_2$ and the matrix. Is ...
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 ...
1
vote
2answers
72 views

Basic set notation - has this textbook made a mistake?

An answer to one question claims that if it is true that $B=\{a,b\}, G=\{\{a,b\}, \{c,2\}\}$, then it is also true that $\{B\}\subseteq G $ But wouldn't $\{B\}=\{\{a,b\}\}$ and there is no ...
0
votes
3answers
56 views

In set notation, why does the meaning of A differ from the meaning of {A}?

If the identity of a set depends on what the set contains, then wouldn't a set represented as A contain the same entities a set represented as {A} contains?
2
votes
1answer
35 views

Is there a name for this structure?

I don't know what branch of math best characterizes what I'm thinking of, so I'll use a concrete example. Suppose you have four boxes, A, B, C, D Box A is an empty box Box B contains Box A ...
1
vote
1answer
59 views

Notation for intersection of two sets each containing a set of intervals

I have a dataset about wireless devices describing when they were connected to a network access point $p$. Any given device can connect to $p$ for some duration, disconnect, and then later reconnect ...
1
vote
2answers
76 views

What does $\alpha+\gamma$ mean when $\alpha$ and $\gamma$ are well-ordered sets?

I was asked to prove the following: let $\gamma$ be a well ordered set with the following property: for any $\alpha$ and $\beta$ well ordered sets, if $\alpha+\gamma=\beta+\gamma$ then ...
0
votes
1answer
68 views

Notation for defining a set of distinct elements.

Suppose I write the following. Let $X = \{x,y,z\}.$ Then its pretty clear that what I really mean is the following. Let $x,y$ and $z$ be fixed but arbitrary; suppose they're distinct; and ...
1
vote
1answer
45 views

Notation for inductive definitions of sets

Is there a formal notation for inductive definitions of sets? For example, like this: $Closure(U,C,A)$ where $U$ is a set, $C$ is a set of constructors (in a simple case, operations on the set $U$), ...
1
vote
1answer
103 views

Notation for the union of a system of sets

The notation $\bigcup_{i \in I}A_{i}$ denotes the union of the range of a function, and as such, is used only if we are considering some function (an indexed family). The generalized associative ...
1
vote
2answers
94 views

Difference between $R^\infty$ and $R^\omega$

I know $R^\omega$ is the set of functions from $\omega$ to $R$. I would think $R^\infty$ as the limit of $R^n$, but isn't that $R^\omega$? The seem to be used differently, but I can't tell exactly ...
1
vote
1answer
61 views

Combining set builder and summation notation

What's the best notation for the sum of a subset? Given $S = \{1,2,3,4,5,6,7\}$, let's say I want to find the sum of the squares of elements less than 4. Initially I used the following notation: ...
0
votes
1answer
25 views

Notation to describe amount of relevant elements in a tuple?

Say, we have the set $A=\{♠,♣,♥,♦\}^3$ and would like to define the following map: \begin{align} f: A &\to \{0,1,2,3\} \\ a &\mapsto \text{amount of ♥'s in the tuple } a \end{align} For ...
0
votes
0answers
22 views

Counting matches in a set

I wish to express the number of elements in a set that meet a certain condition $P$, and I feel silly writing: $X = \{x_1,x_2,x_3...\}$ $\sum\limits_{a\in\{y\in X : P(y)\}}{1}$ Though I do think ...
1
vote
1answer
25 views

How to characterize the set of all real functions defined on $X$.

Let $X$ be an arbitrary set. I consider the set of all real functions defined on $X$. I know that this is usually denoted by $\mathbb{R}^X$. However, I am interested in characterizing each point of ...
1
vote
2answers
74 views

Notation of a function that maps a random element

Let there be a functions $f$ and $g$ such that, $$f:A \times B \mapsto \Re$$ $$g: B \mapsto A$$ where $\forall b \in B$, $g(b)$ is some $a$ such that, $\forall a' \in A, f(a,b) \geq f(a',b)$. (This ...
0
votes
1answer
41 views

The meaning of $Z_a$, where $Z$ is a partitioning of $A$ and $a \in A$

I am unsure about the common usage of subscripting a set with something, more precisely something which might be a member of that set or some other set (as opposed to, say, subscripting a set with an ...
0
votes
1answer
264 views

What the symbol $\subseteq$ represents generally? [duplicate]

My book says that $\subset$ is used to represente any subset, proper or improper, needing in this case to show the anti symmetric property of sets. ($A = B \iff A \subset B \, \, \wedge \,\, B \subset ...
0
votes
1answer
11 views

schematic representation of circular permutation in a set?

I would like to represent digits in a set in a way that the set emphasizes the order of the digits schematically in the set in order to avoid confusion with a set of combinations. For example, how ...
2
votes
1answer
93 views

Difference between $\mathbb Z^+$ and $\mathbb N$

$\mathbb Z^+$ stands for the Positive Integers: $\{1,2,3,4,5\dots\}$ $\mathbb N$ stands for the Natural Numbers: $\{1,2,3,4,5\dots\}$ So what is the difference between $\mathbb Z^+$ and $\mathbb N$? ...
0
votes
2answers
502 views

What is the use of Delta symbol in set theory?

What is the use of $ \Delta $ in set theory?
1
vote
2answers
58 views

Meaning of the set $\mathbb N^\mathbb N$

I came across a question which requires one to check if there's a bijection from the set $ \mathbb N^\mathbb N$ to another set. I've never seen a set defined this way and was wondering if this was ...
2
votes
3answers
33 views

What does $|A|$ denote in set notation?

What does $|A|$ of a set $A$ denote? Also, what does $A\leftrightarrow B$ of sets $A, B$ mean? I encountered this in one of my textbooks which said: Of two sets $A, B$ we know $|B|$ but $|A|$ is ...
0
votes
4answers
115 views

Correct formal interval notation

I can't find any definitive answer on this topic, maybe that's because there isn't one, but I figured if there was a place to ask then SE was it! To describe a set in which $x$ and $y$ are in the ...
1
vote
2answers
45 views

Notation for the Set of All Finite $n$-Tuples from a Set $A$

Let $A \ne \emptyset$. Let $S = \{(a_1, a_2, \ldots , a_n) : a_i \in A$ and $ n \in \mathbb{N}\}$. Now I'm curious if there is a more concise (and standard) way of writing this set down?
1
vote
1answer
205 views

Addition of Sets which isn't union

today a student asked me to prove $${A} \cup B \cup C = A+ B+ C- A\cap B - A\cap C$$ I really had no idea what precisely the "+" sign meant, they insisted, "You know you just add the sets together"; ...