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 and functions, countability and uncountability, etc. More advanced topics should use (set-theory) instead.

This tag is for elementary questions on set theory, focusing on material usually covered in undergraduate set theory texts. More advanced topics should use instead.

Topics include intersections and unions, De Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, etc.

26535 questions
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Is it faster to count to the infinite going one by one or two by two?

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. Skipping half of the number should be really…
Thomas Ayoub
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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 infinities, but they're only words to me. Any help…
Allain Lalonde
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9 answers

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's how you're supposed to do it): I make a…
user1411893
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171
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4 answers

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?
154
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15 answers

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 other two. One can construct each of these…
000
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2 answers

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.
Marso
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120
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6 answers

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 $\limsup A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)$…
Comic Book Guy
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107
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7 answers

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 can anyone produce an explicit formula for such…
Alex Basson
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97
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3 answers

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 classes. What is their relation? Some background…
Asinomás
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90
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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 answers so I will not delete the…
Hugh
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88
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6 answers

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 following form (modified slightly from the…
johne
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6 answers

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 $a$ and $b$, $a R b$ implies $b R a$ by symmetry. Using…
Chao Xu
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86
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7 answers

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 elements in each subset and sending that sum to…
user39794
86
votes
13 answers

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?
ieb
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86
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8 answers

Does mathematics become circular at the bottom? What is at the bottom of mathematics?

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), functions and relations. Now is the "=" taken…
user119615
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