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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Construct a bijection from $\mathbb{R}$ to $\mathbb{R}\setminus S$, where $S$ is countable

Two questions: Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set. $S$ ...
14
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5answers
3k views

How can a set contain itself?

In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this ...
14
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3answers
2k views

How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like?

If we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G_f ...
14
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2answers
2k views

Axiom of Choice Examples

In the wikipedia article, two examples are given which use/ do not use the axiom of choice. They are: Given an infinite pair of socks, one needs AC to pick one sock out of each pair. Given an ...
14
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3answers
2k views

The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable

I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is ...
14
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4answers
3k views

An Intuition to An Inclusion: “Union of Intersections” vs “Intersection of Unions”

Let $E = \{E_k\}_{k \in \mathbb{N}}$ be an infinite sequence of sets. Then, the following inclusion holds: $\bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \quad\subseteq\quad ...
14
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2answers
8k views

Prove that the set of all algebraic numbers is countable

A complex number $z$ is said to be algebraic if there are integers $a_0, ..., a_n$, not all zero, such that $a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n=0$. Prove that the set of all algebraic numbers is ...
13
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7answers
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What are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization?

What are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?
13
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7answers
436 views

Why does $\bigcup_{X \in \mathscr{P}(\mathbb{N})}X = \mathbb{N}$, instead of $\mathscr{P}(\mathbb{N})?$

This is a practice problem I just came across, $$\bigcup_{X \in \mathscr{P}(\mathbb{N})}X =$$ I came up with $\mathscr{P}(\mathbb{N})$, but the book lists $\mathbb{N}$ as the solution. I would've ...
13
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4answers
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Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
13
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3answers
870 views

Sole minimal element: Why not also the minimum?

A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$. The minimum (at most one) of a partially ordered set $S$ is an ...
13
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5answers
7k views

Example of set which contains itself

I am trying to understand Russells's paradox How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself.
13
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4answers
1k views

Proving a set in $\mathbb{Q}$ that is bounded above to have no largest number (greatest element)

I'm studying the set $Q$. At this point I don't know the real numbers. This is the theorem: Let $A$ be the set such that $A=\{p\in \mathbb{Q}^{+}:p^{2}<2\}$. Then $A$ contains no largest ...
13
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2answers
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If A is infinite, does there have to exist a subset of A that is equivalent to A?

I was reading Rudin and there is an alternative definition of infinite set (the first definition is "not finite"): A is infinite if it is equivalent to one of its subset. Then I was wondering the ...
13
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6answers
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A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
13
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3answers
2k views

Proving the countability of algebraic numbers

I am trying to prove that algebraic numbers are countably infinite, and I have a hint to use: after fixing the degree of the polynomial, consider summing the absolute values of its integer ...
13
votes
1answer
387 views

What is wrong with ZFC?

Why are there seemingly so many who want to use ETCS, or HoTT, or similar as a foundation of mathematics? I'm aware that HoTT has a good few good aspects, but that doesn't entirely explain the strong ...
12
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8answers
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Does $\bigcap_{n=1}^{+\infty}(-\frac{1}{n},\frac{1}{n}) = \varnothing$?

When I learn the below theorem: If $I_n$ is closed interval, and $I_{n+1} \subset I_n$, then $$\bigcap I_n \ne \varnothing$$ and someone says if we replace closed interval with open interval, can ...
12
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5answers
2k views

Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?

$\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains ...
12
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8answers
2k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
12
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4answers
2k views

How large is the set of all Turing machines?

How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?
12
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4answers
890 views

Set builder notation: Colon or Vertical Line

I remember once hearing offhandedly that in set builder notation, there was a difference between using a colon versus a vertical line, e.g. $\{x: x \in A\}$ as opposed to $\{x\mid x \in A\}$. I've ...
12
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5answers
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Countable set having uncountably many infinite subsets

Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite ?
12
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4answers
543 views

What is $\bigcup\limits_{n=1}^\infty [0,1-\frac{1}{n}]$?

This is probably a pretty dumb question, but I am confused by set theory again. The question is whether $$\bigcup_{n=1}^\infty \left[0,1-\frac{1}{n}\right]$$ equals $[0,1]$ or $[0,1)$. However, I am ...
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9answers
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Prove that the union of countably many countable sets is countable.

I am doing some homework exercises and stumbled upon this question. I don't know where to start. Prove that the union of countably many countable sets is countable. Just reading it confuses me. ...
12
votes
3answers
488 views

Is every set a subset?

Is every set a subset of a larger set? In other words, for an arbitrary set S, can one always construct a set S' such that S is a proper subset of S'? Is this question even meaningful?
12
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6answers
839 views

Quantifier: “For all sets”

I've seen the following statement a few times: "Let $A$ be a set, then $\emptyset\subseteq A$". Or, written 'more formally': $$ \forall A\,\, \emptyset\subseteq A $$ My doubt is: I've ...
12
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5answers
968 views

Why cannot a set be its own element?

When I study Topology, I met with a problem. On my book, it says 'we cannot admit that there exists a set whose members are all the topological spaces. That will lead to a logical contradiction, that ...
12
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3answers
2k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
12
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1answer
936 views

Do Cantor's Theorem and the Schroder-Bernstein Theorem Contradict?

I am confused as to how Cantor's Theorem and the Schroder-Bernstein Theorem interact. I think I understand the proofs for both theorems, and I agree with both of them. My problem is that I think you ...
12
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4answers
345 views

What explains the asymmetry here?

The image operation distributes over unions: $$f(A \cup B) = f(A) \cup f(B)$$ but about intersections we can only say that $$f(A \cap B) \subseteq f(A) \cap f(B)$$ unless $f$ is injective. Where ...
12
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3answers
678 views

Can there exist an uncountable planar graph?

I'm currently revising a course on graph theory that I took earlier this year. While thinking about planar graphs, I noticed that a finite planar graph corresponds to a (finite) polygonisation of the ...
12
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3answers
3k views

Need help with Recursion Theorem (Set Theory)

The recursion theorem In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, an element $a$ of $X$ and a function $f\colon X \to X$, the theorem ...
12
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2answers
611 views

Hyperreal measure?

If AC be accepted, then there exists a Lebesgue unmeasurable set called Vitali Set. However, I'm curious about measure valued in hyperreal numbers. Argument in disproof of unmeasurability of Vitali ...
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3answers
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Bijection between an open and a closed interval

Recently, I answered to this problem: Given $a<b\in \mathbb{R}$, find explicitly a bijection $f(x)$ from $]a,b[$ to $[a,b]$. using an "iterative construction" (see below the rule). My ...
12
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1answer
902 views

What is the name of the $\in$ symbol and where does it come from?

It looks like a lower-case epsilon, but the Wikipedia page on epsilon states that they are not the same. Does this symbol have a typographic identification outside of mathematics? Where did the ...
12
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1answer
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Why does Cantor's diagonal argument not work for rational numbers?

If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not ...
12
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2answers
2k views

Subset of a finite set is finite

We define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$. How can we prove that a subset of a finite set is finite? It is ...
12
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3answers
346 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
12
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2answers
198 views

A Question regarding disjoint dense sets

If we take the standard topology on $\mathbb{R}$ we can easily find two disjoint sets that are dense, namely $\mathbb{R}\setminus\mathbb{Q}$ and $\mathbb{Q}$. Similarily, if we take the same topology ...
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6answers
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Is there a notation for being “a finite subset of”?

I would gladly use a notation for "A is a finite subset of B", like $$A\sqsubset B \text{ or } A\underset{fin}{\subset} B,$$ but I have never seen a notation for that. Are there any? While ...
11
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7answers
2k views

Is $1$ a subset of $\{1\}$

Is the number $1$ a subset of the set $\{1\}$ just as $\{1\}$ is a subset of the set $\{\{1\}\}$? I'm a little bit confused because $1$ is an element not a ...
11
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5answers
1k views

$A \in B$ vs. $A \subset B$ for proofs

I have to prove a few different statements. The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differs. ...
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5answers
611 views

Prove that every set with more than one element has a permutation without fixed points

I cannot prove this statement so need help. This problem is one of exercises right after the chapter about Hausdorff's maximal principle and Zorn's Lemma. Thus, you cannot use the concept of cardinal ...
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2answers
3k views

Which symbol should be used for an empty set?

Currently, a discussion started on the German Wikipedia article for Empty Set (the German discussion), whether $\emptyset$ or $\varnothing$ should be used or is more common as a symbol for an empty ...
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2answers
1k views

Why is there this strange contradiction between the language of logic and that of set theory?

In standard probability theory events are represented by sets consisting of elementary events. Consider two events for which (as sets) $A \subset B$. If an elementary event $x \in A$ takes places then ...
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5answers
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What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
11
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4answers
794 views

I want to know why $\omega \neq \omega+1$.

In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$. I want to know why $\omega \neq \omega+1$.
11
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4answers
945 views

How does the axiom of regularity forbid self containing sets?

The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem ...
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5answers
999 views

Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...