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
11 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 ...
2
votes
2answers
46 views

A list from each element of another?

Sorry to edit this so much at this late stage, but the question and answers are confused so much by my incorrect use of terminology and the such, I feel that I should clear this up. Where $a$ and ...
0
votes
2answers
61 views

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

I think this is the most succinct answer to the set equinumerosity. $$g(x) = \begin{cases} x & \text{if }x \notin \mathbb{Z},\\ x-1 & \text{if }x \in \mathbb{Z}. \end{cases}$$
0
votes
1answer
28 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 $S= \{1,2,3,4\}.$ ...
-5
votes
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
vote
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 ...
0
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3answers
53 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 ...
0
votes
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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votes
2answers
38 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
26 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 ...
0
votes
2answers
28 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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votes
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 $ $(vA)^c = v(A^c)$ ...
0
votes
1answer
33 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 ...
0
votes
1answer
10 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 ...
2
votes
2answers
50 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\}\\ ...
1
vote
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 ...
0
votes
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
vote
1answer
23 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 ...
0
votes
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
votes
3answers
157 views

Show that a surjective function from $X$ to $J$ does not exist (with a twist!!)

I understand that Suppose that $X$ is a set and $f:X\to \mathcal{P}(X)$ is any function, then $f$ is not surjective. But what if there are two sets $X$ and $Y$, and set of all functions from ...
0
votes
1answer
341 views

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then the union of $A$ and $B$ is a subset of $C$

If $A$ is a subset of $C$ and $B$ is a subset of $C$, then $A\cup B$ is a subset of $C$. I was considering letting $x$ be an element of $A$ and $B$ and going from there, but I'm not sure that that is ...
1
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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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votes
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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votes
2answers
51 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 ...
0
votes
0answers
13 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
votes
1answer
19 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 ...
2
votes
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 ...
0
votes
2answers
107 views

Showing that $A\rightarrowtail A \times \{x\}$ is a bijection

$A\rightarrowtail A \times \{x\}$ where $A$ is any set and $\{x\}$ is an arbitrary one-object set. How would I show the following is a bijection ( one to one and onto)? I know if I turn it into a ...
1
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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?
2
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1answer
44 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 ...
3
votes
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 ...
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 ...
0
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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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votes
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 ≤ ...
1
vote
4answers
66 views

If $f$ is 1-1, prove that $f(A\setminus B) = f(A)\setminus f(B)$

I'm having a tough time with this one. Here's the background: Let $X$ and $Y$ be sets, let $f:X\rightarrow Y$ and let $A,B\subseteq X$. For this proof, we also assume that $f$ is 1-1. I've already ...
0
votes
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 ...
1
vote
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 ...
2
votes
1answer
36 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 ...
1
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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 ...
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votes
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.
0
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2answers
56 views

How to remember various set operations very easily?

I need an way to remember the set operations very easily. Does anybody have any idea? For example, how do you remember the distinction between Set-Intersection and Set-difference? I regularly mess ...
0
votes
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
votes
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 ...
0
votes
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
votes
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,∞)
1
vote
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 ...
0
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2answers
35 views

Countablity of sets

Why do we choose Natural number to describe whether a set is countable or not? How can we say that Natural Number is countable?
1
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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
vote
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 ...