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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5answers
11k 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
votes
3answers
607 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
votes
2answers
936 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
votes
3answers
929 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
257 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
377 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 ...
10
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5answers
2k views

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
347 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
629 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 ...
10
votes
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
821 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
177 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
640 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
votes
1answer
93 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
votes
6answers
260 views

what is a function? please.

Axiom schema of replacement: Let the domain of the function $F$ be the set $A$. Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$) is also a set. — Tarski–Grothendieck ...
9
votes
14answers
2k 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 = ...
9
votes
5answers
933 views

Contradictory definition in set theory book?

I'm using a book that defines $A\setminus B$ (apparently this is also written as $A-B$) as $\{x\mid x\in A,x\not\in B\}$, but then there was an exercise that asked to find $A\setminus A$. Wouldn't it ...
9
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8answers
4k views

What is larger — the set of all positive even numbers, or the set of all positive integers?

We will call the set of all positive even numbers E and the set of all positive integers N. At first glance, it seems obvious ...
9
votes
3answers
737 views

Is $\aleph_0^{\aleph_0}$ smaller than or equal to $2^{\aleph_0}$? [duplicate]

Possible Duplicate: What's the cardinality of all sequences with coefficients in an infinite set? Is $\aleph_0^{\aleph_0}$ smaller than or equal to $2^{\aleph_0}$? I thought I saw this ...
9
votes
5answers
1k views

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 ?
9
votes
4answers
219 views

$\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?

Why does $\bigcup \emptyset = \emptyset$ but $\bigcap \emptyset$ is not defined? If I had to guess, I'd say it's also equal to $\emptyset$.
9
votes
5answers
607 views

If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$? [duplicate]

Prove, disprove, or give a counterexample: If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$. Assume $\mathcal{P}(A)=\mathcal{P}(B)$. Since we know $A \subseteq A$, we know $A \in ...
9
votes
5answers
632 views

Set {1,1} = Set {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\} = ...
9
votes
7answers
2k 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 = ...
9
votes
5answers
446 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 ...
9
votes
4answers
330 views

Infinite sets and their Cardinality

(I am a 13 year old so when you answer please don't use things that are TOO hard even though I actually can understand quite complex stuff) I was studying Infinite sets and their cardinality (not in ...
9
votes
5answers
6k views

How can one prove that the union of two finite sets is again finite, without the use of arithmetic?

The notion that the union of two finite sets is again finite is something I took as intuitively true for quite a while. A proof using arithmetic is relatively straight forward. Suppose $A$ and $B$ ...
9
votes
3answers
764 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
9
votes
2answers
2k views

Cardinality of the infinite sets

Consider the following problem: Which of the following sets has the greatest cardinality? A. ${\mathbb R}$ B. The set of all functions from ${\mathbb Z}$ to ${\mathbb Z}$ C. The ...
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votes
8answers
6k views

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. ...
9
votes
2answers
452 views

Is $\emptyset \in \emptyset$ or $\emptyset \subseteq \emptyset$?

Can someone give an argument, if possible using only the axioms of set theory, because I'm very weak there and have virtually no background, except the usual knowledge of the operation with sets one ...
9
votes
2answers
330 views

How many paths exist between two points in the plane?

Fix distinct $a,b \in \mathbb{R}^2$. In terms of cardinality (say, beth numbers), how many distinct continuous functions $f : [0,1] \rightarrow \mathbb{R}^2$ satisfying $f(0)=a, f(1)=b$ are there? ...
9
votes
7answers
262 views

Is $\mathbb{C}$ equal to $\mathbb{R}^2$?

Complex numbers are usually formally defined as pairs of real numbers. Although there are operations on $\mathbb{C}$, such as complex multiplication, which are not found in operations usually applied ...
9
votes
2answers
526 views

Are known transcendental numbers countable?

Do all known algorithms that generate infinitely many transcendental numbers like Gelfond-Schneider or Liouville only generate countably many? If uncountably many, is this set of measure zero?
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5answers
1k views

Definition of the Infinite Cartesian Product

(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$. (2) On the other hand [Folland, Real Analysis, ...
9
votes
3answers
309 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 ?
9
votes
4answers
253 views

injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto ...
9
votes
4answers
458 views

“The set of all true statements of first order logic”

In one of my lectures, the lecturer put a bunch of examples of sets on the board, stuff like the set of all humans, set of all well typed programs in some programming language, the set of all true ...
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votes
3answers
3k views

The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself - cylinder basis - and it topology

I know the Cantor set probably comes up in homework questions all the time but I didn't find many clues - that I understood at least. I am, for a homework problem, supposed to show that the Cantor ...
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votes
4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
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2answers
2k views

Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?
9
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3answers
328 views

With Choice, is any linearly ordered set well-ordered if no subset has order type $\omega^*$?

I've been fumbling around with order types and ordinals these past few days. I read about partial, total, and well-ordered structures, and I'm curious to see if a linearly ordered set has no subset ...
9
votes
2answers
142 views

Can we expand numbers on the left to produce irrational numbers?

I have seen examples of irrational numbers that are expanded on the right, after the decimal point: e.g. $\pi = 3.14159265...$ But can we expand numbers on the left side as well? e.g. Is ...
9
votes
8answers
281 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
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3answers
3k 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 ...
9
votes
2answers
411 views

The cardinality of Lebesgue sets

Suppose $A=\{S\;|\;S \subset \mathbb R^n, S\text{ is Lebesgue measurable}\}$. What is the cardinality of $A$? Is it the same as the cardinality of all of the real numbers?
9
votes
2answers
661 views

Can a surjection and injection exist but not a bijection? [duplicate]

If I there exists an injection $\phi: S_1 \to S_2$ and a surjection $\tau: S_1 \to S_2$, does there necessarily exist a bijection between sets $S_1$ and $S_2$? I'd like this to be true, but I don't ...
9
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6answers
2k views

Improving my understanding of Cantor's Diagonal Argument

I studied Cantor's Diagonal Argument in school years ago and it's always bothered me (as I'm sure it does many others). In my head I have two counter-arguments to Cantor's Diagonal Argument. I'm not ...
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2answers
3k views

Countable/uncountable basis of vector space

I've stumbled upon this exercise in algebra book, in chapter dealing with vector spaces' dimensions. Prove that basis of the field of real numbers $\mathbb R$ as vector space over the field of ...
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3answers
2k views

Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective

Let $f: A \rightarrow B$ be a function. How can we show that for all subsets $S$ of $A$, $S \subseteq f^{-1}(f(S))$? I think this is a pretty simple problem but I'm new to this so I'm confused. Also, ...