# 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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### Is it faster to count to the infinite going one by one or two by two? [closed]

A child asked me this question yesterday: Would it be faster to count to the infinite going one by one or two by two? And I was split with two answers: In both case it will take an infinite time. ...
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 ...
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'...
38k views

### Are there real-life relations which are symmetric and reflexive but not transitive?

Inspired by Halmos (Naive Set Theory) . . . For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have the ...
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.
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### 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?
4k views

### How do we know that Cantor's diagonalization isn't creating a different decimal of the same number?

Edit: As the comments mention, I misunderstood how to use the diagonalization method. However, the issue I'm trying to understand is a potential problem with diagonalization and it is addressed in the ...
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 ...
6k views

### Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
11k views

### Infinite sets don't exist!?

Has anyone read this article? This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, ...
38k views

8k views

### Why does the Dedekind Cut work well enough to define the Reals?

I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals. I just can't get the ...
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 ...
$\mathbf{Theorem 2.14:}$ Let $A$ be the set of all sequences whose elements are the digits $0$ and $1$. Then A is uncountable, meaning there does not exist a one-to-one mapping of A onto $\mathbb{Z}$. ...
I don’t see the ambiguity that ‘pairwise’ resolves. Surely if $A$, $B$ and $C$ are disjoint sets then they are pairwise disjoint and vice versa? Or am I being dim?