This tag is for elementary questions on set theory - the sort of material covered in "Chapter 0" of graduate texts and in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, ...

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20
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2answers
4k views

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

Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
17
votes
7answers
6k 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 ...
40
votes
5answers
6k views

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

How do I 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 ...
13
votes
3answers
1k 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
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?
34
votes
6answers
5k 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 ...
10
votes
1answer
3k 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?
4
votes
2answers
3k 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 these ...
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 ?
25
votes
6answers
4k 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 ...
29
votes
5answers
12k 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 ...
15
votes
3answers
1k 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 ...
25
votes
6answers
4k 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?
3
votes
3answers
469 views

Prove $f(S \cup T) = f(S) \cup f(T)$

$f(S \cup T) = f(S) \cup f(T)$ f(S) encompasses all x that is in S f(T) encompasses all x that is in T thus the domain being the same, both the LHS and RHS map to the same ys, since the function ...
12
votes
2answers
10k 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 ...
17
votes
2answers
853 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 ...
17
votes
3answers
4k 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 ...
4
votes
2answers
1k 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 ...
28
votes
3answers
5k views

difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
15
votes
8answers
2k views

Why Are the Reals Uncountable?

Let us start by clarifying this a bit. I am aware of some proofs that irrationals/reals are uncountable. My issue comes by way of some properties of the reals. These issues can be summed up by the ...
10
votes
3answers
3k 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 ...
20
votes
4answers
726 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 ...
2
votes
3answers
2k 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 ...
6
votes
3answers
671 views

How is $\epsilon_0$ countable?

In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is countable. Thanks.
2
votes
2answers
642 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 ...
10
votes
4answers
557 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 ...
8
votes
4answers
853 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 ...
6
votes
3answers
637 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
3answers
3k 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 ...
5
votes
2answers
2k views

Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$

I am working on this exercise for an introductory Real Analysis course: Show that |$\mathbb{R}$| = |$\mathbb{R}^2$|. I know that $\mathbb{R}$ is uncountable. I also know that two sets $A$ and ...
18
votes
1answer
688 views

Characterising functions $f$ that can be written as $f = g \circ g$?

I'd like to characterise the functions that ‘have square roots’ in the function composition sense. That is, can a given function $f$ be written as $f = g \circ g$ (where $\circ$ is function ...
4
votes
4answers
388 views

Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality

Let $f: A\longrightarrow B$ be a function. 1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$. 2)Give an example of a function $f$, and sets ...
4
votes
2answers
2k 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 ...
12
votes
10answers
3k views

Intuitive explanation for how could there be “more” irrational numbers than rational?

I've been told that the rational numbers from zero to one for a countable infinity, while the irrational ones form an uncountable infinity, which is in some sense "larger". But how could that be? ...
9
votes
3answers
1k 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, ...
3
votes
4answers
2k 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 $, ...
18
votes
2answers
1k views

Infinite product of measurable spaces

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining product ...
12
votes
4answers
3k 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?
15
votes
1answer
2k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
11
votes
3answers
1k 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 $Y$. ...
8
votes
7answers
4k 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 ...
12
votes
2answers
1k 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.
11
votes
5answers
430 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 ...
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 ...
8
votes
4answers
520 views

Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$. Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$ Therefore, $y\in F(A)$ and ...
20
votes
3answers
2k 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 ...
7
votes
3answers
697 views

Bijecting a countably infinite set $S$ and its cartesian product $S \times S$

From Herstein's Topics in Algebra (exercise 14, section 1.2): If $S$ is infinite and can be brought into one-to-one correspondence with the set of integers, prove that there is one-to-one ...
6
votes
2answers
506 views

How to show equinumerosity of the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?

How to show equinumerosity of the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic? Not homework, practice exercise.
4
votes
3answers
957 views

Surjectivity of Composition of Surjective Functions

Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto?
3
votes
3answers
223 views

Prove or find a counter example to the claim that for all sets A,B,C if A ∩ B = B ∩ C = A ∩ C = Ø then A∩B∩C ≠ Ø

This is a homework question of mine that Ive answered and I'd really like some feedback on my solution. Prove or find a counter example to the claim that for all sets $A,B,C$, $$\text{if }A\cap ...