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, ...

learn more… | top users | synonyms

12
votes
2answers
202 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 ...
11
votes
14answers
3k views

Using set notation, define the set of even natural numbers between 100 and 500.

Using set notation, define the set of even natural numbers between 100 and 500. This is what I have so far: $P$ is even numbers so the set of natural numbers between 100 and 500 would be $$P = ...
11
votes
6answers
1k views

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
votes
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
votes
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. ...
11
votes
5answers
637 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 ...
11
votes
2answers
4k 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 ...
11
votes
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 ...
11
votes
5answers
1k views

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
votes
4answers
803 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
votes
3answers
1k views

Why is cardinality of set of even numbers = set of whole numbers?

I recently watched a YouTube video on Banach-Tarski theorem (or, paradox). In it, the presenter builds the proof of the theorem on the basis of a non-intuitve assertion that there as as many even ...
11
votes
4answers
1k 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 ...
11
votes
4answers
5k views

Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
11
votes
4answers
504 views

Looking for a problem where one could use a cardinality argument to find a solution.

I would like to find an exercise of the type: Find some $x$ in $A\setminus B$. Solution: since $A$ is uncountable and $B$ is countable such $x$ exists...
11
votes
1answer
327 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 ...
11
votes
5answers
1k 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
votes
4answers
4k views

Empty intersection and empty union

If $A_\alpha$ are subsets of a set $S$ then $\bigcup_{\alpha \in I}A_\alpha$ = "all $x \in S$ so that $x$ is in at least one $A_\alpha$" $\bigcap_{\alpha \in I} A_\alpha$ = "all $x \in S$ so that ...
11
votes
6answers
893 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 ...
11
votes
3answers
3k views

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 ...
11
votes
5answers
15k views

Is the void set (∅) a proper subset of every set ?

I am a bit confused about the concept of proper subsets,precisely whether to include one or both of the void set and the set itself. An extract from my module goes like this : Obviously,every set is ...
11
votes
3answers
679 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.
11
votes
4answers
364 views

A “Cantor-Schroder-Bernstein” theorem for partially-ordered-sets

If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?
11
votes
4answers
3k 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 ...
11
votes
2answers
268 views

Characterization properties of number sets $\mathbb{N},\mathbb{ Z},\mathbb{Q},\mathbb{R},\mathbb{C}$

When people say that a structure is defined up to isomorphism means, accordingly, that they assume certain properties that make it completely determined under certain operations and relations. ...
11
votes
1answer
674 views

Why is the collection of all groups considered a proper class rather than a set?

According to Wikipedia, The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and ...
11
votes
2answers
440 views

Why doesn't this work imply that there are countably many subsets of the naturals?

Cantor's theorem shows us that the power set of the natural numbers is uncountably infinite. But today (and before remembering Cantor's proof) I was trying to prove the incorrect version: that the ...
11
votes
4answers
2k 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 ...
11
votes
3answers
6k views

A finite set always has a maximum and a minimum.

I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated.
11
votes
3answers
4k views

limit inferior and superior for sets vs real numbers

I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...
11
votes
2answers
87 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 ...
11
votes
3answers
503 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 ...
11
votes
1answer
227 views

How to formulate the P v.s. NP problem as a formal statement inside the language of set theory?

I've read a lot that some computer scientists believe that P v.s. NP could turn out to be independent of ZFC. The thing that puzzled me is how to formulate this inside the language of set theory? I ...
11
votes
4answers
2k views

Is there a shorthand notation for adding an element to a set?

I know that if you want to refer to the set $ A $ with the element $ x $ added, you can write $ A \cup \{x\} $. But is there a common shorthand for this?
11
votes
1answer
15k views

Mathematical notation for the maximum of a set of function values

I have a question about the proper notation of the following (simplified) example: I want to express that I have a value alpha, which is the maximum of a set of n values. Each value in the set is the ...
11
votes
4answers
200 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 ...
11
votes
2answers
268 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
votes
1answer
212 views

Is there any connection between the symbol $\supset$ when it means implication and its meaning as superset? [duplicate]

A rather old-fashioned symbol for logical implication is $\supset$ (see list of logic symbols). For example $p \supset q$ means $p \implies q$ or $p \rightarrow q$ in more recent notations. Is there ...
11
votes
1answer
156 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 $f(a)=f(b)=f(a+b)$.
11
votes
2answers
175 views

Showing that $|A\cap B|/|A \cup B| + |B\cap C|/|B \cup C| - |A\cap C|/|A \cup C| \leq 1$ for finite sets $A,B,C$.

If $A$, $B$ and $C$ are finite sets, prove that $$ \frac{|A\cap B|}{|A \cup B|} + \frac{|B\cap C|}{|B \cup C|} - \frac{|A\cap C|}{|A \cup C|} \leq 1. $$ It seem's simple, but I tried it for ...
11
votes
1answer
177 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
11
votes
0answers
207 views

Can $[0,1]$ be partitioned into an uncountable union of uncountable sets? [duplicate]

I was thinking about this: $[0,1]$ can be partitioned into a countable union of uncountable sets. Write $[0,1]=(0,1]\cup \{0\}:$ $$(0,1]=\bigcup_{n=1}^{\infty}\Big(\frac{1}{n+1},\frac{1}{n}\Big]$$ ...
10
votes
9answers
5k views

Is the sum of all natural numbers countable?

I do not even know if the question makes sense. The point is rather simply. If I have the sum of all natural numbers, $$\sum_{n\in \mathbb{N}}n$$ this is clearly "equal to infinity". But since ...
10
votes
6answers
4k views

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. ...
10
votes
5answers
788 views

$\{1,1\}=\{1\}$, origin of this convention

Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements? Like $\{1,1,2,3\} = ...
10
votes
1answer
642 views

Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
10
votes
7answers
3k views

Is an empty set equal to another empty set? [duplicate]

I have a definition that claims that two sets are equal A = B, if and only if: $\forall x ( x \in A \leftrightarrow x \in B)$ An empty set contains no elements. If I define the sets: A = ...
10
votes
1answer
619 views

Is directed set countable, if for each element there are only finitely many smaller ones?

A directed set is a pair $(A,\leq)$ where $\leq$ is a reflexive, transitive relation such that for any $x,y\in A$ we have some $z$ such that $x,y\leq z$. (This comes up when dealing with categorical ...
10
votes
4answers
2k views

Is any relation which contains only one ordered pair transitive?

I need clarification. Let $A=\{1,2,3\}$ be a set and $R=\{(1,2)\}$ be a relation on $A$. Is it a Transitive relation? I am confused because some text books say $R$ is transitive if it contains only ...
10
votes
5answers
494 views

What is the canonical definition of an open set?

The definition of an open set that I see in most topology texts(like the ones found in Topology by Munkres and another w/ the same title by Hocking & Young, or Basic Topology by Armstrong) is that ...
10
votes
2answers
1k views

countable group, uncountably many distinct subgroup?

I need to know whether the following statement is true or false? Every countable group $G$ has only countably many distinct subgroups. I have not gotten any counter example to disprove the statement ...