Questions tagged [elementary-set-theory]

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, differences and complements, De Morgan's laws, Venn diagrams, relations, etc. More advanced topics should use (set-theory) instead.

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144
votes
8answers
35k views

How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
112
votes
2answers
26k views

Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ [closed]

Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
68
votes
7answers
36k views

Show that the set of all finite subsets of $\mathbb{N}$ is countable.

Show that the set of all finite subsets of $\mathbb{N}$ is countable. I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the ...
94
votes
4answers
30k 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?
64
votes
6answers
34k views

Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
65
votes
8answers
7k views

Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign)...
151
votes
9answers
8k views

What Does it Really Mean to Have Different Kinds of Infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable ...
54
votes
7answers
23k views

Cardinality of set of real continuous functions

The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?
13
votes
3answers
11k views

Cartesian Product of Two Countable Sets is Countable [closed]

How can I prove that the Cartesian product of two countable sets is also countable?
10
votes
3answers
4k views

Associativity of symmetric difference of sets

By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. ...
15
votes
6answers
4k views

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

Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
29
votes
7answers
14k 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 $...
16
votes
7answers
11k views

How to prove that $\mathbb{Q}$ ( the rationals) is a countable set

I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it ...
67
votes
6answers
20k views

How does Cantor's diagonal argument work?

I'm having trouble understanding Cantor's diagonal argument. Specifically, I do not understand how it proves that something is "uncountable". My understanding of the argument is that it takes the ...
52
votes
5answers
44k views

Proof that the irrational numbers are uncountable

Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the ...
71
votes
6answers
38k views

lim sup and lim inf of sequence of sets.

I was wondering if someone would be so kind to provide a very simple explanation of lim sup and lim inf of s sequence of sets. For a sequence of subsets $A_n$ of a set $X$, the $\limsup A_n= \bigcap_{...
76
votes
3answers
21k views

What are the differences between class, set, family, and collection?

In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about ...
26
votes
4answers
12k 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 ...
6
votes
3answers
4k views

Infinite DeMorgan laws

Let $X$ be a set and $\{Y_\alpha\}$ is infinite system of some subsets of $X$. Is it true that: $$\bigcup_\alpha(X\setminus Y_\alpha)=X\setminus\bigcap_\alpha Y_\alpha,$$ $$\bigcap_\alpha(X\setminus ...
9
votes
4answers
2k views

What's the cardinality of all sequences with coefficients in an infinite set?

My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
63
votes
6answers
14k views

Why isn't reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ ...
17
votes
4answers
13k 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 ...
29
votes
2answers
23k 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 ...
33
votes
4answers
1k views

Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?

I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of ...
25
votes
7answers
5k views

Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized

Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$. I know that the number of elements of the power set of $A$ is $2^n$, i.e. $|\mathcal{P}(A)|=2^n$. I came across ...
9
votes
4answers
7k views

Proving the Cantor Pairing Function Bijective

How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$ How would ...
6
votes
2answers
5k views

An infinite subset of a countable set is countable

In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I ...
38
votes
7answers
55k 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. ...
44
votes
7answers
12k views

In set theory, how are real numbers represented as sets?

In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets? Thanks for the answers. Several answers ...
59
votes
10answers
46k views

Is there a bijective map from $(0,1)$ to $\mathbb{R}$?

I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
31
votes
2answers
6k views

Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
6
votes
4answers
5k views

Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $

I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated! Let $A$ be any uncountable set, and let $B$ be a countable ...
37
votes
6answers
3k views

When does the set enter set theory?

I wonder about the foundations of set theory and my question can be stated in some related forms: If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not ...
35
votes
2answers
8k views

Why doesn't Cantor's diagonal argument also apply to natural numbers?

In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string. My question is: why can't we begin by representing each natural ...
22
votes
2answers
3k views

How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?

One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$. What is the proof of this fact? As Arturo pointed out in his comment, in computer science this ...
33
votes
3answers
8k views

Proof that the real numbers are countable: Help with why this is wrong

I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The ...
29
votes
2answers
10k views

The cardinality of the set of all finite subsets of an infinite set

Let $X$ be an infinite set of cardinality $|X|$, and let $S$ be the set of all finite subests of $X$. How can we show that Card($S$)$=|X|$? Can anyone help, please?
19
votes
3answers
3k views

The Aleph numbers and infinity in calculus.

I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus? Are they interchangeable/transposable in ...
6
votes
6answers
1k views

Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$

For any pair of sets $A$ and $B$, we can define $A\le B$ iff there exists injection $f\colon A\rightarrow B$. I am trying prove that $$(A\le B)\vee (B\le A).$$ I have tried assuming $\neg (A\le B)$,...
9
votes
7answers
9k views

Partition of N into infinite number of infinite disjoint sets? [duplicate]

Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
7
votes
3answers
1k views

bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate]

I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I can'...
10
votes
4answers
14k views

Empty set does not belong to empty set

Herbert in his book "Elements of set theory" on page no 3 says that we can form the set $ \{ \emptyset \} $ whose only member is $\emptyset $. Note that $ \{ \emptyset \} \neq \emptyset $, ...
30
votes
4answers
6k views

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 $Y$. ...
6
votes
2answers
7k views

countably infinite union of countably infinite sets is countable

How do you prove that any collection of sets $\{X_n : n \in \mathbb{N}\}$ such that for every $n \in \mathbb{N}$ the set $X_n$ is equinumerous to the set of natural numbers, then the union of all ...
8
votes
5answers
2k views

$A \oplus B = A \oplus C$ imply $B = C$?

I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both. But when it comes to solving ...
38
votes
3answers
61k views

Is the power set of the natural numbers countable?

Some explanations: A set S is countable if there exists an injective function $f$ from $S$ to the natural numbers ($f:S \rightarrow \mathbb{N}$). $\{1,2,3,4\}, \mathbb{N},\mathbb{Z}, \mathbb{Q}$ are ...
40
votes
4answers
15k views

Cardinality of the set of all real functions of real variable

How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?
17
votes
4answers
14k views

Understanding equivalence class, equivalence relation, partition

I'm 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 ...
12
votes
3answers
14k views

The set of real numbers and power set of the natural numbers

I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these ...
29
votes
5answers
16k 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.