This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, ...

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1answer
20 views

Symbol clarification

Okay, so I've read a few different meanings for the exclamation point in a statement. For example: $$!\exists x \in O \ni 2x < 5$$ The only question I have is about the Exclamation point in front ...
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1answer
15 views

Defining a relation on a set with conditions

Define a relation R on R (All Real Numbers) as follows: For all real numbers x and y mTn if and only if 3 | (m - n). I'm not sure what the vertical bar here means. Normally it means "such as" but ...
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1answer
32 views

Defining A Binary Relation On All Real Numbers

Define a relation R on $\mathbb R$ (Set of all Real Numbers) as follows: For all real numbers $x$ and $y$, $x \mathrel{R} y$ if and only if $x = y$. Since the set of all real numbers is infinite, how ...
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2answers
15 views

Relation between successor cardinals and power sets

What are the known relation between successor cardinals $\kappa^+$ and power sets $2^\kappa$ (when GCH is not assumed)? For example, is it true that $\kappa^+ \le 2^\kappa \le \kappa^{++}$? In ...
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1answer
30 views

no. of disordered pairs of disjoint subsets

I found this question in a book. The same question has been asked before, but I want a more generalised and rigorous, so to speak, answer. The question reads- " Consider the set $S= \{1,2,3,4\}.$ ...
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2answers
64 views

Prove $(0,\infty)$ is equinumerous to $[0, \infty)$.

I think this is the most succinct answer to the set equinumerosity. $$g(x) = \begin{cases} x & \text{if }x \notin \mathbb{Z},\\ x-1 & \text{if }x \in \mathbb{Z}. \end{cases}$$
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1answer
42 views

does the empty set = infinity? [on hold]

I need note two distinctions prior to asking if 'n' is an bounded variable. If, taking for instance a secondhand-function (sf) to be something contained within a container, and a firsthand-function ...
1
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1answer
12 views

Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...
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3answers
54 views

If $A \cap B \cap C = \varnothing$, is one $A \cap B$, $B \cap C$ or $C \cap A$ empty too? [duplicate]

How do I give counter example to this? Prove or find a counter example to the following claim: For all sets $A$, $B$, $C$ if $A\cap B\cap C=\varnothing$, then either $A\cap B=\varnothing$ or ...
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2answers
41 views

Is the null set $\emptyset$ a real subset of any set?

My query is simple. If $A=\{1,2,3\}$. the subsets of $A$ are $\{1,2,3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1\},\{2\}, \{3\}, \{\}$. As per the textbook, the subset $\{1,2,3\}$ is not a real subset of $A$ ...
1
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1answer
29 views

transitive closure and number of elements in relation?

I see an example as follows: in relation $R=\{(a,b), (b,c), (b,d), (c,e), (d,e), (c,f), (e,a) \}$, on set $\{a,b,c,d,e,f\}$. we have $30$ elements in the transitive closure of $R$. How number of ...
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2answers
29 views

P vs NP and Countable vs Uncountable Decision Space

I have noticed that whenever the scope of a problem is pushed to infinity, problems in NP have an uncountably infinite decision space whereas problems in P seem to have a countably infinite decision ...
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1answer
46 views

Mathematical Relations [on hold]

For $v,z\in\mathbb R$, $A\subseteq\mathbb R$ define $vA = \{va :a\in A\}$ and $A+z=\{a+z :a \in A\}$. Prove that: $v(A\cap B) = vA \cap vB$ $v(A\cup B) = vA \cup vB $ $(vA)^c = v(A^c)$ ...
1
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1answer
30 views

Defining sets as countable and infinite

Which of the following sets are finite? countably infinite? uncountable? (Be careful -- don't apply theorems for finite sets to infinite sets and don't apply theorems for countable sets to uncountable ...
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1answer
21 views

Countably Infinite Collections of Sets

I need to find examples of: (a) A countably infinite collection of pairwise disjoint finite sets whose union is countably infinite (b) A countably infinite collection of nonempty sets whose union is ...
1
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1answer
24 views

$\bigcup \alpha$ where $\alpha$ is a finite ordinal.

Given a finite ordinal, is it correct in saying $\bigcup \alpha = \alpha - 1$? As an illustrative example consider $3 = \{\emptyset , \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$. I believe ...
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0answers
19 views

How to write the family of sets whose elements are the sets in a sequence of sets

I am wondering, given a sequence of sets $( X_n )$, how do we write the corresponding family of sets whose elements are the sets in the sequence? Of course, the same question applies to nets as well. ...
0
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1answer
11 views

Constructing an almost contained set from a family of sets with strong finite intersection property.

I don't even know if this is true but I have a feeling I've read it's true somewhere. A counterexample or a proof would be equally welcome, or a link to where I can find more information. (Maybe the ...
0
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1answer
35 views

How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
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2answers
60 views

Proving $A$ is a subset of $B$

I'm trying to understand the proof behind showing a set is a subset of another set, but I'm struggle to do so. Can some one help using this example to show: $A \subseteq B$? Here $A = \{x | x = 4n ...
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1answer
21 views

What is the limit of the cardinality of a set of bins in finite range, as bin width approaches zero?

Let's say that we divide the region $(0,1)$ into $N$ bins of width $1/N$. Of course, it makes sense to take the limit $1/N \rightarrow 0$ in this configuration, because that's simply how we define an ...
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0answers
14 views

Cantor Bendixson rank of a Cartesian product

I am trying to find where the proof of the following equality was published. I $CB(X \times Y) = CB(X) \oplus CB(Y)$ where CB represents the Cantor-Bendixson rank of a set and $\oplus$ is the ...
2
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1answer
20 views

How many subsets of $S$ are there that contain $x$ but do not contain $y$?

Let $S$ be a set of size $37$, and let $x$ and $y$ be two distinct elements of $S$. How many subsets of $S$ are there that contain $x$ but do not contain $y$? This question is on a practice exam ...
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2answers
52 views

Power Set Of a Complement of an Infinite Set?

In order to find a Power Set of (B \ A), an infinite Set, would you keep finding elements until both sets have one in common? For example: $$\begin{align} A &= \{x \mid x = 2n, n \in \mathbb ...
2
votes
2answers
52 views

Calculate Intersection with a Non Finite Set?

What is the best way to answer Intersection or Union based questions with a set that is not finite? such as this: Calculate: $A \cap B$ $$\begin{align} A&=\{x\mid x=n+9, n\in\mathbb N\}\\ ...
2
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2answers
35 views

If a function fg is surjective under composition and f is surjective, is g surjective?

If a function $fg$ is surjective under composition and $f$ is surjective, is $g$ surjective? I think not, since $f$ could be a many to one function and $g$ could send elements only once to elements ...
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2answers
29 views

How do I show that the following map establishes a bijection between $\mathbb Q$ and $\mathbb Z \times \mathbb{Z}_{>0}$

Define $$f:\mathbb Z \times \mathbb{Z}_{>0} \to \mathbb{Q}$$ by $$f(p, q) = \frac{p}{q} $$ Edit: Is there an explicit map then that is a bijection?
3
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0answers
36 views

Initial and final topologies

Suppose that $X_i$ are topological spaces, and $X_i \xrightarrow{f_i} Y$ are a family of maps into the set $Y$. The final topology on $Y$ is defined to be the finest topology on $Y$ such that each ...
3
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0answers
51 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
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2answers
36 views

question about proving subset inclusion

normally when one proves subset inclusion, one usually take any $x$ from the subset, and proves that it is also in the superset. e.g. Set $A=$ all triangles Set $B=$ all shapes with a sum of of ...
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1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
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1answer
33 views

Decide whether set is convex, connect and bounded.

Let $A=\{ \left(x,y,z \right)\in \mathbb{R}^3 : x^2+y^2-z^2+1<0\}$. Decide whether set A is: a) convex (definition i know: Set $A\in \mathbb{R}^k$ is convex set if for all $x,y \in A$ line segment ...
2
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1answer
37 views

Does set theory help understand machine learning or make new machine learning algorithms?

When I was in a university, I didn't major in math but took some math classes. However, I dropped out of math classes pretty quick. Some person recommended that I learn some set theory because it'll ...
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1answer
33 views

Operations no Sets

Let $A$ be the set $\{x : x \in \mathbb Z \ \text {and either} \ x ≤ −2 \ \text {or} \ x ≥ 5 \}$ and let $B$ be the set $\{ −3, −2, −1, 4, 5, 6, 7 \}$. Find the following : $A\cup B = \{x : - x ≤ ...
0
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4answers
118 views

Is $ (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)$ true for all sets $A, B, C$ and $D$?

Is $(A \times B) \cup (C \times D) = (A \cup C) \times (B \cup D)$ true for all sets $A, B, C$ and $ D?$ I tried to wrap my head around this, but I have absolutely no idea what is going on here. How ...
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1answer
51 views

Are intersection of power set and power set of intersection equal? [duplicate]

Is $P(A) ∩ P(B) = P(A ∩ B)$? At first glance it seems like its not true. I tried writing out all the values of the power set using examples but I'm not sure on how to prove it.
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2answers
35 views

Prove if the following is true or provide a counterexample if it is not

For all sets A and B, |P(A × B)| $\ne$ |P(A) × P(B)| My first instinct is that it is false and I picked sets like A = {1}, B = {2} but when you write out the power set of these sets you end up with ...
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3answers
30 views

Is the following set operation true?

Prove the following or else find a counter example: For all sets $A$, $B$, and $C$, $$((A \cup B) − C) \cup (A \cap B) = ((A − B) \cup (B − A)) − C$$ For the life of me, I can't figure out if its ...
2
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1answer
15 views

The inverse image of the image of $X$

I'm working on some exercises in Bert Mendelson's Introduction to Topology book in the first chapter and there's this question about functions: If $f:A\rightarrow B$ is injective, then for every ...
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1answer
34 views

Is $\mathbb{R}^{a\text{ x } a}$ equivalent to $\mathbb{R}^{a^2}$?

In my linear algebra course I was given $\mathbb{R}^{a\text{ x }a} $ as "the set of all $a$ by $a$ matrices". While $\mathbb{R}^{a^2}$ was "the set of vectors with $a^2$ coordinates". Ex ...
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0answers
22 views

Proving finite/infinite sets

For j$\in\mathbb{Z}^+$, let $A_j$$\subseteq$$\{$1,..., j$\}$. Suppose that for some n$\in$$\mathbb{Z}^+$, we have B$\subseteq$$\cup^{1}_{j=1}$$A_j$. Is B necessarily finite? Prove it or give a ...
0
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1answer
23 views

Find a one-to-one correspondence (i.e, a bijection) [on hold]

Find a bijection between the following sets where {[]} denotes a closed interval and {()} denotes an open interval A = [-3,7] and B = [41,100] & A = (-∞,-3) and B = (8,∞)
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1answer
24 views

Is $\operatorname{card}(I)=\operatorname{card}(D)$

When I was answering number of integrable functions is greater than number of differentiable functions I got to wonder if the inequality was strict. So with $\mathcal I$ being the set of integrable ...
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2answers
18 views

Prove that the greatest lower bound of $F$ (in the subset partial order) is $\cap F$.

This is one of the question I'm working on: Suppose $A$ is a set, $F \subseteq \mathbb{P(A)}$, and $F \neq \emptyset$. Then prove that the greatest lower bound of $F$ (in the subset partial ...
1
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1answer
32 views

Sum of two dedekind cut is a cut

Given $A_1,A_2\in\mathbb R$, define the following: $$ A_1+A_2= \{x + y: x \in A_1, y \in A_2\} $$ I was able to prove that it is not equal to $\mathbb Q$ and isn't the empty set and but I can't prove ...
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2answers
40 views

What is a good free software to draw complicated Venn diagrams?

The important feature I want is this : I would like to draw two sets as say ovals in solid line but I would like to have the border line in some neighborhood of their two intersection points to be ...
2
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1answer
45 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...
1
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1answer
52 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
1
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2answers
70 views

Recursively defining sets of strings discrete math

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set K? Can someone ...
0
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1answer
51 views

Consider $f : \mathbb{N} \to (-1,1) \cap \mathbb{Q}, \ n \overset{f}{\mapsto} \sin(n)$. Is $f$ a surjection? If not can we make it a surjection?

This is not homework! I have recently been thinking about the properties of the sine function and whether it can effectively map elements from certain spaces to entirely "fill out" another space. I ...