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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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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1answer
497 views

Any branch of math can be expressed within set theory, is the reverse true?

Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property? I am asking ...
12
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3answers
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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 ...
11
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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
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5answers
456 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
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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
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4answers
2k views

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
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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 ...
11
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3answers
581 views

Simple cardinal arithmetic

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
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3answers
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Preimage of generated $\sigma$-algebra

For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$. Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
11
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4answers
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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 ...
11
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3answers
256 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 ...
11
votes
1answer
201 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, ...
11
votes
2answers
164 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
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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 ...
11
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3answers
258 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
122 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
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0answers
196 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
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9answers
4k 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
5answers
1k 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 ...
10
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7answers
5k views

difference between maximal element and greatest element

I know that it's very elementary question but I still don't fully understand difference between maximal element and greatest element. If it's possible, please explain to me this difference with some ...
10
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2answers
2k 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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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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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 ...
10
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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 ...
10
votes
1answer
533 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
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4answers
760 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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4answers
420 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...
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6answers
8k 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 ...
10
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3answers
247 views

Are the error terms of the partial sums of inclusion-exclusion unimodal?

I often teach inclusion-exclusion: $$|A ∪ B| = |A| + |B| − |A ∩ B|$$ by suggesting that $|A∩B|$ is a correction factor for $|A|+|B|$. Then I teach the three set version: $$|A∪B∪C| = |A| + |B| + |C| ...
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5answers
746 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 ...
10
votes
1answer
280 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 ...
10
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6answers
310 views

Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false? [duplicate]

There exists an $X\in A$ such that $P(X)$. When $A$ is the empty set then this statement is false because there is nothing in $A$ that when plugged in for $X$, makes $P(X)$ come out True. However, ...
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5answers
10k 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 ...
10
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3answers
542 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 ...
10
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2answers
861 views

Is there an empty set in the complement of an empty set?

Currently taking a logic class and trying to understand this. You have two set $A$ and $B$. Both sets are empty sets. Is set $A$ a subset of the complement of set $B$? Assume the context is the ...
10
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3answers
881 views

Cofinality and its Consequences

(1)In set theory, what is the purpose for defining the concept of cofinality?is it that important? (2)The concept of cofinality finally leads to 2 types of infinite cardinal, for which the first ...
10
votes
2answers
255 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. ...
10
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2answers
372 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 ...
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5answers
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What are some examples of classes that are not sets?

After reading about Russell's paradox, I see that the set of all sets does not exist, so instead it is called a class. What other commonly known classes exist that are not sets? I know the class of ...
10
votes
4answers
330 views

An uncountable linearly independent set

I've been taking a course in linear algebra and one of the first things we defined was linear independence. It made me wonder how big a linearly independent set can be, in particular whether we can ...
10
votes
2answers
612 views

Set theory puzzles - chess players and mathematicians

I'm looking at "Basic Set Theory" by A. Shen. The very first 2 problems are: 1) can the oldest mathematician among chess players and the oldest chess player among mathematicians be 2 different ...
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4answers
842 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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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 ...
10
votes
5answers
783 views

The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$

What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection ...
10
votes
2answers
1k 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 ...
10
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2answers
173 views

Bolzano-Weierstrass for sequences of sets

Let $\mathcal{A}_n,\,n\in\mathbb{N}$ be a sequence of subsets of, say, $\mathbb{R}$. Let $\limsup_{n\rightarrow\infty} \mathcal{A}_n = \{x:x\in\mathcal{A}_n\mbox{ for infinitely many } n\}$, and ...
10
votes
4answers
601 views

How does one get the formula for this bijection from $\mathbb{N}\times\mathbb{N}$ onto $\mathbb{N}$?

When showing that $\mathbb{N}\times\mathbb{N}$ is in bijection with $\mathbb{N}$, it seems standard to give a proof by picture that shows a way to systematically weave through all the points in ...
10
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1answer
84 views

a totally ordered set with small well ordered set has to be small?

doing something quite different the following question came to me: 1)If you have a totally ordered set A such that all the well ordered subset are at most countable, is it true that A has at most the ...
10
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2answers
4k 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 ...