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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556 views

Set theory puzzles - chess players and mathematicians

I'm looking at "Basic Set Theory" by A. Shen. The very first 2 problems are: 1) can the oldest mathematician among chess players and the oldest chess player among mathematicians be 2 different ...
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4answers
507 views

Difference between a function and a graph of a function?

Formally, I learned that a function $f: X \to Y$ is a subset $f \subset X \times Y$ subject to the condition that for every $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. We write ...
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votes
8answers
251 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
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votes
2answers
925 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 ...
10
votes
5answers
667 views

The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$

What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection ...
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4answers
2k views

Need help with Recursion Theorem (Set Theory)

The recursion theorem In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, an element $a$ of $X$ and a function $f\colon X \to X$, the theorem ...
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votes
1answer
387 views

Any branch of math can be expressed within set theory, is the reverse true?

Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property? I am asking ...
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2answers
167 views

Bolzano-Weierstrass for sequences of sets

Let $\mathcal{A}_n,\,n\in\mathbb{N}$ be a sequence of subsets of, say, $\mathbb{R}$. Let $\limsup_{n\rightarrow\infty} \mathcal{A}_n = \{x:x\in\mathcal{A}_n\mbox{ for infinitely many } n\}$, and ...
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4answers
510 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 ...
10
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2answers
142 views

A Question regarding disjoint dense sets

If we take the standard topology on $\mathbb{R}$ we can easily find two disjoint sets that are dense, namely $\mathbb{R}\setminus\mathbb{Q}$ and $\mathbb{Q}$. Similarily, if we take the same topology ...
10
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1answer
82 views

a totally ordered set with small well ordered set has to be small?

doing something quite different the following question came to me: 1)If you have a totally ordered set A such that all the well ordered subset are at most countable, is it true that A has at most the ...
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1answer
97 views

Coloring of positive integers

Suppose $f:\mathbb{Z}^+\longrightarrow X$ is a function, with $X$ a finite set. Is it true that there are $a,b\in\mathbb{Z}^+$ such that $f(a)=f(b)=f(a+b)$.
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6answers
240 views

what is a function? please.

Axiom schema of replacement: Let the domain of the function $F$ be the set $A$. Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$) is also a set. — Tarski–Grothendieck ...
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votes
5answers
1k views

Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?

$\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains ...
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8answers
3k views

What is larger — the set of all positive even numbers, or the set of all positive integers?

We will call the set of all positive even numbers E and the set of all positive integers N. At first glance, it seems obvious ...
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votes
1answer
451 views

Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
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5answers
539 views

Set {1,1} = Set {1}, origin of this convention

Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements? Like $\{1,1,2,3\} = ...
9
votes
5answers
383 views

What is the canonical definition of an open set?

The definition of an open set that I see in most topology texts(like the ones found in Topology by Munkres and another w/ the same title by Hocking & Young, or Basic Topology by Armstrong) is that ...
9
votes
4answers
287 views

Infinite sets and their Cardinality

(I am a 13 year old so when you answer please don't use things that are TOO hard even though I actually can understand quite complex stuff) I was studying Infinite sets and their cardinality (not in ...
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votes
5answers
694 views

Naive set theory question on “=”

So I picked up a couple of good undergraduate-level books over the weekend and have been working through them... In Algebra: Chapter 0, the author of the text writes: The prototype of the ...
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votes
3answers
226 views

Are the error terms of the partial sums of inclusion-exclusion unimodal?

I often teach inclusion-exclusion: $$|A ∪ B| = |A| + |B| − |A ∩ B|$$ by suggesting that $|A∩B|$ is a correction factor for $|A|+|B|$. Then I teach the three set version: $$|A∪B∪C| = |A| + |B| + |C| ...
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6answers
285 views

Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false? [duplicate]

There exists an $X\in A$ such that $P(X)$. When $A$ is the empty set then this statement is false because there is nothing in $A$ that when plugged in for $X$, makes $P(X)$ come out True. However, ...
9
votes
2answers
470 views

Are known transcendental numbers countable?

Do all known algorithms that generate infinitely many transcendental numbers like Gelfond-Schneider or Liouville only generate countably many? If uncountably many, is this set of measure zero?
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4answers
1k views

Formal proof for A subset of the real numbers, well ordered with the normal order of $\mathbb R$, is at most $\aleph_0$

I tried to write a formal proof for the theorem: $A$ subset of $\mathbb R$ well ordered by the normal order $\implies A$ is at most of cardinality $\aleph_0$. Any suggestions? Thanks.
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3answers
340 views

How does the axiom of regularity forbid self containing sets?

The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem ...
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2answers
1k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
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3answers
264 views

A “Cantor-Schroder-Bernstein” theorem for partially-ordered-sets

If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?
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4answers
238 views

injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$

Today a friend of mine told me a nice fact, but we couldn't prove it. The fact is that there is an injection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ defined by the fomula $(m,n)\mapsto ...
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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?
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4answers
992 views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
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votes
1answer
751 views

What is the name of the $\in$ symbol and where does it come from?

It looks like a lower-case epsilon, but the Wikipedia page on epsilon states that they are not the same. Does this symbol have a typographic identification outside of mathematics? Where did the ...
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6answers
1k views

Improving my understanding of Cantor's Diagonal Argument

I studied Cantor's Diagonal Argument in school years ago and it's always bothered me (as I'm sure it does many others). In my head I have two counter-arguments to Cantor's Diagonal Argument. I'm not ...
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votes
2answers
407 views

Can a surjection and injection exist but not a bijection? [duplicate]

If I there exists an injection $\phi: S_1 \to S_2$ and a surjection $\tau: S_1 \to S_2$, does there necessarily exist a bijection between sets $S_1$ and $S_2$? I'd like this to be true, but I don't ...
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3answers
2k 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 ...
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votes
3answers
254 views

Why is this proof of $\mathbb{N}\times\mathbb{N}$ being countable not formal?

My copy of Introduction to Real Analysis: Bartle and Sherbert gives: Theorem: The set $\mathbb{N}\times\mathbb{N}$ is countable. Informal Proof: Recall that $\mathbb{N}\times\mathbb{N}$ ...
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votes
3answers
249 views

Intersection of two 'huge' sets in the plane

Consider two sets on the plane $A=\mathbb{Q}\times \mathbb{R}$ and $B=\mathbb{R}\times \mathbb{Q}$. We know that $A\cap B=\mathbb{Q}\times \mathbb{Q}\neq\emptyset$. What about the general cases? That ...
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3answers
260 views

Question about members in sets

Let $A_1,A_2,...,A_n$ be sets with $k$ members in $A_i$ for every $1\le i\le n$. Suppose that the $A_i$ satisfy: 1) $|A_i\cap A_j| = 1$ for all $i\ne j$, 2) $A_1\cap A_2\cdots\cap A_n =\emptyset$. ...
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votes
4answers
340 views

A problem about symmetric relations on finite sets.

We have these assumptions: $X$ is a finite set. $\sim$ is an irreflexive symmetric relation on $X$. for any subset $Y\subseteq X$ we define $$\mathcal{Cl}(Y)=\{A\subseteq Y\mid(\forall a,b\in ...
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votes
2answers
141 views

Question from 'How to Prove It'

Below is the question from the book mentioned above: Suppose $f : A \rightarrow B$ and $R$ is an equivalence relation on $A$. We will say that $f$ is compatible with $R$ if $∀x \in A\forall y ∈ ...
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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 ...
8
votes
4answers
1k views

Why is an empty function considered a function?

A function by definition is a set of ordered pairs, and also according the Kurastowski, an ordered pair $(x,y)$ is defined to be $$\{\{x\}, \{x,y\}\}.$$ Given $A\neq \varnothing$, and ...
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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 ?
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4answers
419 views

Is it correct to say that $\mathbb{R}$ has fewer elements than $\mathbb{C}$ if both are infinite?

My math teacher said that. I disagreed, but he said that I was wrong. But I'm not convinced - is it really right? Please notice that I'm not talking about $\mathbb{R}$ $⊂$ $\mathbb{C}$, but ...
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3answers
944 views

The largest number system

If my number set construction memory doesn't fail me (I'll edit if errors are pointed out), we start out with Peano's axioms to get to $\mathbb{N}$, and in the need of an additive inverse for its ...
8
votes
4answers
186 views

$\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?

Why does $\bigcup \emptyset = \emptyset$ but $\bigcap \emptyset$ is not defined? If I had to guess, I'd say it's also equal to $\emptyset$.
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7answers
4k views

Example of Partial Order that's not a Total Order and why?

I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial ...
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votes
6answers
361 views

Why should $|2^\mathbb{N}|>|\mathbb{N}^2|$ be true?

I've been thinking a bit about infinite things lately, and this question I had wondered about came back to me. One of the classic expository demonstrations of Cantor's work is the two equally ...
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votes
4answers
779 views

Is any relation which contains only one ordered pair transitive?

I need clarification. Let $A=\{1,2,3\}$ be a set and $R=\{(1,2)\}$ be a relation on $A$. Is it a Transitive relation? I am confused because some text books say $R$ is transitive if it contains only ...
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votes
3answers
491 views

Hilbert Hotel: what if countably many buses each with countably many guests arrived?

Situation: There's a hotel owner David Hilbert who owns a hotel with countably many (infinity that can be mapped by natural number surjectively) rooms, and there are countable guests who lived inside ...
8
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1answer
763 views

function that is its own inverse

$f(f(x))=x \ \forall x \in \mathbb{R}$. I am trying to prove there exists an irrational $t$ such that $f(t)$ is also irrational. I have been trying things like assume $t$ irrational implies $f(t)$ is ...