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, (un)...

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9
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0answers
434 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
7
votes
0answers
285 views

Cardinality of set of groups

After analyzing this question I started wondering. Thoughts. Everyone can give a simple example of a countable group $G, |G| = \aleph_0$ which has uncountable number $2^{\aleph_0}$ subgroups, for ...
7
votes
0answers
157 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\...
6
votes
0answers
54 views

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
6
votes
0answers
251 views

Is there really anything wrong with Bourbaki's Set Theory?

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time ...
6
votes
0answers
143 views

Tool that draws Venn diagram from subset relation

Is there a tool (LaTeX, JavaScript, Mathematica..) that allows one to draw Venn diagram automatically from subsets relations, e.g. $$A\subset A+B$$ $$A\subset C$$ $$C\subset C+D$$ $$B \not\subset ...
6
votes
0answers
213 views

Validity of my proof for a Cartesian Product in Tao's book

Before all, I'm apologize if my question is too common here. I'm only want to know if my proof is correct or need some adjustments. I'm reading the Terence Tao's Analysis book as I mentioned in my ...
5
votes
0answers
203 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
5
votes
0answers
146 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 ...
4
votes
0answers
94 views

Every family $\mathscr{A} $ of sets satisfies $|\mathscr{A} \setminus \mathscr{A}| \geq |\mathscr{A}|$

Let $\mathscr{A} $ be a set of sets. Let's denote $\{A \setminus B : A,B \in \mathscr{A}\}$ by $\mathscr{A} \setminus \mathscr{A} $. The Marica-Schönheim theorem in combinatorics says that $|\...
4
votes
0answers
63 views

Deducing the existence of particular functions $\mathbb{N}\longrightarrow\mathbb{Q}$ in the context of Tom Leinster's “Rethinking Set Theory”

This question concerns the set theory given by Tom Leinster in his paper "Rethinking Set Theory," available here: http://arxiv.org/abs/1212.6543 In this paper, axioms for set theory are given in the ...
4
votes
0answers
118 views

How can numbers be used in mathematics that do not belong to a countable set?

The following question distinguishes between mathematical objects including real numbers that can be used in mathematical discourse and such which cannot. Definitions of the former can be sent from a ...
4
votes
0answers
68 views

Number of members of a set in geometry

Suppose we have a $2n \times 2n$ grid of unit squares divided into $2 \times 2$ contiguous squares. Suppose we wish to have a set of these unit squares such that At least one unit square of each 2 x ...
4
votes
0answers
71 views

Proving there is no injection from $V^V$ to $V$

Let $V$ be a set such that $V$ is not a singleton. I have to prove that there doesn't exist an injection from $V^V$ to $V$. As predecessor to this exercise, I have proven that there doesn't exist a ...
4
votes
0answers
110 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
4
votes
0answers
263 views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that $\...
4
votes
0answers
145 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 $...
4
votes
0answers
129 views

Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
4
votes
0answers
406 views

Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
4
votes
0answers
166 views

Prove $f_\infty: A_\infty \rightarrow B_\infty$ is a bijection

Update: I was given some hints at how to approach this problem $A_\infty $ and $B_\infty$ are sets, not maps. (which is strange because there are function definitions coming into play here) The ...
4
votes
0answers
102 views

Linear dimension of banach spaces

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $||x||=|x_1|+...+|x_n|$. ...
4
votes
0answers
54 views

Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
4
votes
0answers
349 views

Crititism of the set-theoretic definition of natural numbers

A while ago I read in a book (or a paper?) that a very well-known mathematician (Saunders Maclane?) in his lectures used to mock the classical set-theoretical definition of natural numbers: 0 = {}, 1 ...
4
votes
0answers
90 views

The Theory of Probabilistic Sets

${P}_m := \{ \Phi_{1m}, \Phi_{2m}, \ldots , \Phi_{nm} \}$ where $$|\Phi_{im} \rangle = \wp_{1m} | \phi_1 \rangle + \wp_{2m} | \phi_{2m} \rangle + \cdots + \wp_{km} | \phi_{km} \rangle$$ such that $\...
4
votes
0answers
98 views

$\left(0,1\right]\neq\biguplus_{k=1}^{n}\bigcap_{j=1}^{k}G_{j},$ Proving elegantly

How can I show (without making many distinctions by cases) that the equality $$ \left(0,1\right]=\biguplus_{k=1}^{n}\bigcap_{j=1}^{k}G_{j}, $$ can't hold, if $G_{j}\in\mathcal{A}\cup\left\{ M^{c}\...
4
votes
0answers
405 views

Sum of bijective functions

Can anyone please help me with this? Let $f,g_{1}, g_{2},\ldots,g_{k} \in \mathbb{Q}^{\mathbb{N}}$. $f$ is a sum of $g_{1},g_{2},\ldots,g_{k}$ if for every natural number $n$ $$f(n) = g_{1}(n) +g_{...
3
votes
0answers
45 views

My proof that $f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2)$

I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ...
3
votes
0answers
37 views

Restriction to equivalence relation is equivalence relation

Let $\mathcal{R}$ be relation on $A$ and $A_0 \subseteq A$. The $\mathbf{restriction}$ of $\mathcal{R}$ to $A_0$ is defined to be the relation $\mathcal{R} \cap (A_0 \times A_0) $. $\mathbf{...
3
votes
0answers
25 views

Let $A$ be a set with $m$ elements and let $B$ be a set with $n$ elements where $m,n\in \omega$ and $m>n$. If $f:A\to B$, then $f$ is not injective

So I am still learning how to work with infinite sets, and this particular problem is giving me some issues. Right now, I am trying to pick some $x_1,x_2\in A$ such that $f(x_1) = f(x_2)$ to serve as ...
3
votes
0answers
59 views

Definition of Ordered Pair

This is probably a more open ended question. Kuratowski's definition of ordered pair is that $(a,b) = \{\{a\},\{a,b\}\}$. This is basically just subset of a power set. But say we have a probability ...
3
votes
0answers
45 views

Some Trouble Understanding set theory

I'm currently in a discrete mathematics class and we've recently been discussing set theory. I feel like I have basic understanding of how to actually prove set relations when a question asks to do so....
3
votes
0answers
62 views

Bijective continuous map between $\mathbb{R}$ and $\mathbb{R}^2$

I am currently attending a course on point-set topology and fundamental group. Today we proved in class that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic with their normal Hausdorff topologies....
3
votes
0answers
57 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
3
votes
0answers
25 views

Cartesian Product of a Union and an Intersection

I am given the Cartesian Product of an equation: $(A \cap C) \times (B \cup A)$ As being $\{(5,1),(5,4),(6,1),(6,4)\}$ And the sets: $B=\{1,9,4\}$ and $C=\{5,6,7,8\}$ And so I figure that $(A \...
3
votes
0answers
23 views

Clarification about a metric

This is rather an easy question but I am a bit confused about the following metric $\rho(\mathcal{G},\mathcal{H}) := \sup_{A\in \mathcal{G}} \inf_{B\in \mathcal{H}} \mu(A \triangle B) + \sup_{B\in \...
3
votes
0answers
95 views

Question on infinite T4 topological space

I have this question here from a course in general topology course I could not answer all of so I am asking here, it reads: Let $ (X,\tau) $ be a $ T_4 $ topological space (normal Hausdorff) with $...
3
votes
0answers
39 views

Behavior of the null set?

I am trying to work out problems to understand the null set beyond the notion of, "an empty box". Therefore, I have tried to work out some problems concerning set operations involving the null set. I ...
3
votes
0answers
184 views

Is There a Common Definition of “Finite Support”?

I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term (...
3
votes
0answers
70 views

How can functions be written as ordered triples?

In a function $ \langle f,A,B \rangle $, I know that the domain $ A $ and the co-domain $ B $ are not restricted to being sets. They can be proper classes. In that case, how can we write functions as ...
3
votes
0answers
73 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
3
votes
0answers
281 views

Is this a way to construct mathematics?(logic vs. set theory)

I recently asked a question about the fact that logic and set theory seems circular. link I got a lot of good and thoughtful answers, that probably explains everything, but I must admit I did not ...
3
votes
0answers
72 views

Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A \rightarrow B$ is injective then $|A| \leq |B|$ and if $g: B \rightarrow A$ is injective then $|B| \leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, ...
3
votes
0answers
47 views

How to generate families of sets without replacement symmetry?

I need to generate a family of sets with as few symmetries as possible. If convenient, let the size of each elements of the family be a fixed $s$ and the number of elements in the family be a fixed $n$...
3
votes
0answers
52 views

Writing some basic sentences in the language of set theory.

I am having some trouble proving that some basic sentences in the language $L_\in$ of first order set theory are $\Sigma_1$ or $\Pi_1$, the reason being that I do not know how exactly to write them. ...
3
votes
0answers
55 views

A reflexive relation?

Suppose I have a set $\mathrm{A}=\{1,2,3,4,5\}$ and a reflexive relation $\mathrm R$ defined on $\mathrm A$, i.e., $$R\colon A\mapsto A\quad\textrm{and}\quad \mathrm R\textrm{ is reflexive.}$$ Is ...
3
votes
0answers
42 views

Proving Equivalence Relation $xPy$ iff $ y = x + n\pi$ on The Reals

I am trying to prove that the relation P on $\mathbb{R}$ by the rule $\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$ From what I can ...
3
votes
0answers
62 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
votes
0answers
84 views

Why is this not a poset after adding zero?

The problem    Consider the following set for divisibility. {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 96}. If 0 is added, the divisibility relation set will no longer be a poset. Please ...
3
votes
0answers
114 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $...
3
votes
0answers
104 views

Is it possible to create division via Set Theory?

I've been reading a book on Set Theory (Charles C. Pinter), and it says, ...set theory is recognized to be the cornerstone of the "new" mathematics... [emph. added] and that ...we can still ...