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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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 ...
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4answers
2k 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 ...
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2answers
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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 ...
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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7answers
1k 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 ...
12
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4answers
532 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 ...
12
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5answers
540 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 ...
12
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3answers
447 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?
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4answers
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Interpretation of limsup-liminf of sets

What is an intuitive interpretation of the 'events' $$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$ and $$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$ when $A_n$ are ...
12
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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 ...
12
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3answers
664 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 ...
12
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5answers
777 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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1answer
878 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
343 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 ...
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4answers
5k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
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2answers
530 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 ...
12
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4answers
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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1answer
863 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
2k views

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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3answers
1k 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 ...
12
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3answers
293 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
181 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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9answers
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What is an Empty set?

We define the term "Set" as, A set is a collection of objects. And an "Empty set" as, An empty set is a set which contains nothing. First problem I encountered: How the definition of ...
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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 ...
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6answers
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Why is the supremum of the empty set $-\infty$ and the infimum $\infty$? [duplicate]

I read in a paper on set theory that the supremum and the infimum of the empty set are defined as $\sup(\{\})=-\infty$ and $\inf(\{\})=\infty$. But intuitively I can't figure out why that is the case. ...
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2answers
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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
779 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 $\omega \neq \omega+1$.
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6answers
10k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
11
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4answers
676 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
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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.
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5answers
889 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 $a \sim b$ ...
11
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1answer
309 views

In naive set theory ∅ = {∅} = {{∅}}?

In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly. ∅ is an empty set, so having an empty set as an element of a set that ...
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6answers
821 views

Naive set theory question on “=”

So I picked up a couple of good undergraduate-level books over the weekend and have been working through them... In Algebra: Chapter 0, the author of the text writes: The prototype of the ...
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3answers
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Cardinality of the set of prime numbers

It was proved by Euclid that there are infinitely many primes. But what is the cardinality of the set of prime numbers ? Cantor showed that the sets $\mathbb{Q}$ and $\mathbb{Z}$ have the same ...
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4answers
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Formal proof for A subset of the real numbers, well ordered with the normal order of $\mathbb R$, is at most $\aleph_0$

I tried to write a formal proof for the theorem: $A$ subset of $\mathbb R$ well ordered by the normal order $\implies A$ is at most of cardinality $\aleph_0$. Any suggestions? Thanks.
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3answers
621 views

How to prove that from “Every infinite cardinal satisfies $a^2=a$” we can prove that $b+c=bc$ for any two infinite cardinals $b,c$?

Prove that if $a^2=a$ for each infinite cardinal $a$ then $b + c = bc$ for any two infinite cardinals $b,c$. I tried $b+c=(b+c)^2=b^2+2bc+c^2=b+2bc+c$, but then I'm stuck there.
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3answers
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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 $xRz$, ...
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4answers
750 views

When do two functions become equal?

When do two functions become equal? I have stumbled over this definition of equality of functions in elementary real analysis. Let $X$ and $Y$ be two sets. Let $f:X\rightarrow Y$ and ...
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3answers
4k views

Bijection from $\mathbb R$ to $\mathbb {R^N}$

How does one create an explicit bijection from the reals to the set of all sequences of reals? I know how to make a bijection from $\mathbb R$ to $\mathbb {R \times R}$. I have an idea but I am not ...
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4answers
1k views

Difference between a function and a graph of a function?

Formally, I learned that a function $f: X \to Y$ is a subset $f \subset X \times Y$ subject to the condition that for every $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. We write ...
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2answers
78 views

Does there exist a function $g\in \mathbb{N}^\mathbb{N}$ s.t. $\{f\mid f\circ f=g\}$ is not empty and finite?

I'm struggling with this question and can't figure it out. The question was too long for the title so I will write it once more: Does there exist a function $g : \mathbb{N} \longrightarrow ...
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2answers
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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 ...
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3answers
459 views

Why is the collection of all algebraic extensions of F not a set?

When proving that every field has an algebraic closure, you have to be careful. In this article https://proofwiki.org/wiki/Field_has_Algebraic_Closure, and as I have been told on this site, if we have ...
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3answers
4k views

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 ...
11
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1answer
206 views

What does it mean for a set to have “structure”?

I understand that a set is like a list of things, except that the order doesn't matter and that you can't have any duplicates in a set. For example: $\{3, 1, 4, 2\}$ is the same set as $\{1, 2, 3, ...
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4answers
199 views

How find this minimum of the value $f(1)+f(2)+\cdots+f(100)$

Give the positive integer set $A=\{1,2,3,\cdots,100\}$, and define function $f:A\to A$ and (1):such for any $1\le i\le 99$,have $$|f(i)-f(i+1)|\le 1$$ (2): for any $1\le i\le 100$,have ...
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3answers
260 views

Problem about subsets of $\{1, 2,\dots,n\}$

Let $A=\{1, 2,\dots,n\}$ What is the maximum possible number of subsets of $A$ with the property that any two of them have exactly one element in common ? I strongly suspect the answer is $n$, but ...
11
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1answer
134 views

Coloring of positive integers

Suppose $f:\mathbb{Z}^+\longrightarrow X$ is a function, with $X$ a finite set. Is it true that there are $a,b\in\mathbb{Z}^+$ such that $f(a)=f(b)=f(a+b)$.