# 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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### 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'...
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.
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 ...
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?
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 ...
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)...
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 ...
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?
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?
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 $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. ...
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?
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### 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 ...
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 ...
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### 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?
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'...
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 $\{ \emptyset \} \neq \emptyset$, ...
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 $Y$. ...
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 ...
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 ...
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 ...
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)?
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 ...