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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9
votes
0answers
412 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 ...
8
votes
0answers
128 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
7
votes
0answers
271 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
154 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
242 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
140 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
211 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
199 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
144 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
61 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
106 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
65 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
67 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
106 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
223 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
129 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
126 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
356 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
164 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
101 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
53 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
341 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\{ ...
4
votes
0answers
398 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) ...
3
votes
0answers
24 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
56 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
44 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 ...
3
votes
0answers
59 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 ...
3
votes
0answers
50 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
22 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
81 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
37 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
170 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
70 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
253 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
70 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
46 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 ...
3
votes
0answers
50 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
53 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
58 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
113 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
96 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 ...
3
votes
0answers
83 views

an introduction to axiomatic set theory that is not enderton

So I've been reading Endertons Elements of set theory which is easy to understand when it comes to the essential axioms, but there are a number of topics which seem to gloss over important ...
3
votes
0answers
49 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
3
votes
0answers
35 views

What is the product of an empty family of similiar algebras, that is $\prod\langle \mathbf{A}_i \ | \ i \in I \rangle $, where $I = \emptyset$?

What is the product of an empty family of similiar algebras, that is $\prod\langle \mathbf{A}_i \mid i \in I \rangle $, where $I = \emptyset$? The family $\langle \mathbf{A}_i \mid i \in \emptyset ...