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, de Morgan's laws, Venn diagrams, relations, functions, ...

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3
votes
3answers
57 views

Finding a bijective function between an open disk and the open square

How can I find a bijective function between these two sets? $$\{(x,y)\in\mathbb{R}^2 \,|\, x^2+y^2<1\}, \quad (-1,1) \times (-1,1) .$$ I already thought of first writing between 2nd and set ...
1
vote
2answers
28 views

How do I find the type of relation on an infinite set?

Imagine I'm given a set A = {∅, {∅}, {{∅}}, {{{∅}}},…} where ∅ is empty-set. Then I also have a relation on that set (actually on its power set) defined as: $R \subseteq \wp(A) \times \wp(A)$, where ...
1
vote
1answer
23 views

Find bijective correspondence between the sets

Find bijective correspondence between the set of all functions of $X$ in the set $\left\{ 0,1 \right\}$ and the power set of set $X$ and find $| 2 ^ X |$, if $| X | = n.$ My thoughts: ...
0
votes
1answer
39 views

How to show venn diagram?

How to show the following sets by Venn diagrams? Case 1: $$A=\{1,2,B\},B =\{3,4\}$$ Case 2: $$A=\{1,2,3,4\}, B=\{3,4\}$$
-2
votes
1answer
42 views

How many elements are in the set $S^S$, where $S=\{a,b\}$?

If set $S =\{a,b\}$, then how many elements will be in set $S^S$? Here $S^S$ is {Set S is Exponent of S}. Do we need to do cross product like $S*S$ when it says $(S^S)$. Please advise.
1
vote
2answers
33 views

Proof in set theory

Let $A,B,C$ -- subsets in some fixed set. Prove that $A \cap B \subseteq C$ iff $A \subseteq \overline{B} \cup C$. Have no ideas how to prove this. On the language of definitions we have $$x ...
0
votes
0answers
29 views

How do I prove this assertion? [duplicate]

Let $A$ be countable union of countably infinite sets. Then $A$ is also countable.
-1
votes
2answers
31 views

Sets cardinality definition

I have a question about cardinality definition. How can we formally define cardinality for finite set using only maps from natural numbers to the set? UPD One says that the cardinality can be ...
1
vote
2answers
25 views

A simple way to know whether a well-ordered set has a subset of a certain type

Following my last question, Does $\Bbb R-\Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$, I would like to have better tools to look at a set and know what order types can it have. I ...
3
votes
1answer
35 views

Cardinality of all linear transformations from $\Bbb R^3$ to $\Bbb R^2$

I tried to calculate the cardinality of all linear transformations from $\Bbb R ^3$ to $\Bbb R^2$. This is my answer- I would like to know how to formalize it better. A transformation is defined in ...
3
votes
1answer
57 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
0
votes
1answer
34 views

Problems with this Cartesian Product definition

Supposed I do not define ordered pair in the usual Kuratowski way $(x,y) = \{\{x\},\{x,y\}\}$. I left the ordered pair undefined but with the propriety $(x,y) = (x',y') \iff x=x'\text{ and }y=y'$. ...
3
votes
1answer
55 views

Does $\Bbb R-\Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$

Does $\Bbb R- \Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$? I thought of taking the subset to be A={$n\cdot \sqrt{m}:n\in\Bbb N,m\in P$} where P is the set of all prime numbers, ...
1
vote
1answer
56 views

Cantor-Bernstein Proof

Currently, I am studying Set Theory, and have come to the point of proving the Cantor-Bernstein Theorem (if $|A| \leq |B|$ and $|B| \leq |A|$, then $|A| = |B|$). Now, I am studying from Jech and ...
4
votes
2answers
102 views

Is the cardinality of $\mathbb{Z^R}$=$\mathbb{R^Z}$?

Previously in this question, we have found that $\mathbb{R^Z}$ is uncountable and its multiset of components, denoted by $$K = \{ (..., 0, 0, w, 0, 0, ... ) : w \in \mathbb{R} \}$$ where for each ...
2
votes
2answers
65 views

Can we define $ℝ^A$ where A is uncountable?

The question is pretty straightforward. How can we define the expression $ℝ^A$ when $A$ is an uncountable set? For example what is defined by forms such as $ℝ^ℝ$ or $ℝ^ℂ$? If $A$ is countable,then ...
1
vote
0answers
42 views

How to translate set propositions involving power sets and cartesian products, into first-order logic statements?

As seen from an earlier question of mine one can translate between set algebra and logic, as long as they speak about elements (a named set A is the same as {x ∣ x ∈ A}). However I've stumbled upon ...
0
votes
2answers
23 views

partial functions basics

$f: \mathbb{Z} \to\mathbb{N}$ is defined as $$ f(x)= \begin{cases} 2x-1, & \text{$x \gt 0$} \\ -2x, & \text{$x \le 0$} \end{cases} $$ one to one proof f is onto proof ...
-1
votes
1answer
29 views

Functions from $\{w,x,y,z\}$ to $\{a,b,c\}$

I'm having some problems understanding how functions and Big-O notation works... I've checked a couple of other threads here but still unsure Let's say I have $A = \{w, x, y, z\}$ and $B = \{a, b, ...
0
votes
0answers
36 views

What is the correct mathematical notation for a finite set?

My random variable $y$ belongs to a finite set of real numbers. I am writing a document and I need to write something like $y\in \mathbb{R}$. what should I put next to $\mathbb{R}$ which shows that ...
2
votes
0answers
53 views

Set Theory Notation: What does it mean to “\” one set with another? [duplicate]

What does the "\" operator mean in the above context?
2
votes
3answers
32 views

Elementary set theory notation verification

Reading Velleman's "How To Prove It" I came across the following expression: $$ x \in\bigcup\{\mathscr P(A)\mid A\in \mathcal F\} $$ such that $\mathcal F$ is a family of sets, $A$ is a set, and ...
2
votes
1answer
24 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
1
vote
1answer
25 views

Bringing ordinals to standard polynomial form

By the definition of multiplication and addition of ordinals, the following rules follow- Multiplication- $n\cdot\omega=\omega$ while $\omega\cdot n > \omega$ Addition- $n+\omega=\omega$ while ...
0
votes
4answers
27 views

what does the union of those 3 events imply

Let there be 3 events: A=a dish got broken B= electric product stopped working C= the car got broken Write the following event, D= at least 2 problem occurred. $D=(A\cap ...
1
vote
2answers
50 views

Can I prove set propositions using first-order logic?

I'm studying logic and sets and I have to say there's a strong similarity between the two. Most boolean/logic axioms also apply to sets. At the end of my course I also studied first-order logic (or ...
3
votes
3answers
113 views

(Is it a set?) Set of all months having more than 28 days.

Set is a well defined collection of distinct objects. Is the following is a set? Set of all months having more than 28 days. I'm confused here. Because on one hand I think that it is well ...
2
votes
1answer
22 views

The least $\aleph$ that has no surjective map from $m$ to it.

Without $AC$. Let $\aleph^*(m)$ be the least aleph that $\not\leq^* m$. How to show that $\aleph^*(m)$ exists and $\aleph^*(m)= \{\alpha\in ON\mid\ \alpha\leq^*m\}$. $ON$ is the class of all ...
1
vote
1answer
27 views

Notation question $|X^2|$

I am studying a little bit of set theory, and one of the questions in the book (in Efe A. Ok's real analysis book) asked to show that $\dim(X,\succeq)\leq |X^2|$, where $X$ is a finite set and ...
1
vote
4answers
90 views

$A=\{A,\emptyset\}$ and axiom of regularity

The axiom of regularity says: (R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$. From (R) it follows that there is no infinite membership chain (imc). Consider this set: ...
28
votes
8answers
2k views

Are there fewer positive integers than all integers? [duplicate]

In our 6th grade math class we got introduced to the concept of integers. With all the talk about positive and negative, it got me wondering. Is the amount of elements in $\mathbb{Z^+}$ less than the ...
3
votes
3answers
60 views

Number of subsets

Let $|X|=n$. How to find all number of subsets $X$ consisting of an even number of elements?
1
vote
2answers
24 views

Show that $\bigcup_{n\geq 1} A_n\subset B$

I have a rahter silly question.. but am a bit unsure nonetheless. If I have to show that $\bigcup_{n\geq 1} A_n\subset B$ is it enough to show that there is ONE $A_n$ with $A_n\subset B$?
0
votes
0answers
19 views

Symmetric Difference between sets [on hold]

Let $A$ and $B$ two finite sets and consider their symmetric difference $A \bigtriangleup B$. It's clear that $|A \bigtriangleup B|=1$ if and only if $A \subsetneqq B$ and $|B \setminus A|=1$ (or ...
3
votes
0answers
72 views

Is this a way to construct mathematics?(logic vs. set theory)

I recently asked a question about the fact that logic and set theory seems circular. link I got a lot of good and thoughtful answers, that probably explains everything, but I must admit I did not ...
1
vote
3answers
26 views

How do I prove propositions involving power sets and cartesian products?

In an earlier question I asked regarding how to solve specific propositions involving set unions/intersections etc. What helped greatly is the use of axioms and rules that I could use to prove the ...
3
votes
1answer
34 views

Reference request for Zermelo's construction of natural numbers

I have heard that back in 1908 Zermelo proposed to use $\emptyset,\{\emptyset\},\{\{\emptyset\}\},\ldots$ as the natural numbers. However, later von Neumann proposed the alternative approach of ...
3
votes
3answers
73 views

Sets $A_1,A_2,A_3,…$ with $\dots\in A_3\in A_2\in A_1$

In $\textsf{ZFC}$, is there a sequence of sets $A_1,A_2,A_3, \dots$ such that $$\cdots \in A_3\in A_2\in A_1?$$
2
votes
1answer
25 views

Question about countable and uncountable map correspodence

So I was solving the following question: If $B$ is uncountable with countable subset $A\subset B$, prove that there exists a one-to-one correspondence between $B$ and $B-A$. So here is how I ...
0
votes
2answers
71 views

Which is the cardinality of the set of all functions $\mathbb{R} \to \mathbb{R}$? [duplicate]

Which is the cardinality of the set of all functions $\mathbb{R} \to \mathbb{R}$ ? For the relations in $\mathbb{R}$ I think the cardinality is the cardinality of $\mathcal{P(\mathbb{R^2})}$ , but ...
2
votes
2answers
30 views

Show that these three statements are logically equivalent.

show that these three statements are logically equivalent. $A \subseteq B, A \cup B = B, A \cap B = A$. I am unsure how to begin this, so i have set up as follows First I must show that. $A ...
1
vote
1answer
35 views

Let $A$ be a set and $x>0$ integer. What is $x^A$?

Let $A$ be a set and $x>0$ integer. What is $x^A$? or can we define such a set? As an example: $2^A$ is power set of $A$. What is $3^A$ or any $x^A$?
1
vote
1answer
41 views

$f$ is continuous $\iff f(\bar A) \subset \overline{f(A)}$

The problem is: $f:X\to Y$: any map. $f$ is continuous $\iff \forall A\subset X, \ f(\bar A) \subset \overline{f(A)}$ My understanding is: Suppose $f$ is continuous. $\forall A\subset X, A ...
0
votes
3answers
66 views

How to solve set problems using algebra?

I'm a beginner math student and I'm studying sets. I've studied some things relating sets and am now trying to solve my first exercises. The first of which have to do with understanding whether ...
2
votes
1answer
13 views

Prove $\{A_1, \cdots ,A_n\}$ is a partition of $A$ given $\{S_1, \cdots, S_n\}$ is a partition of $\Omega$

Let $A \subseteq \Omega$ and $\{S_1,\cdots ,S_n\}$ be a partition of $\Omega$. Let $A_i = A \cap S_i$. Prove $\{A_1, \cdots ,A_n\}$ is a partition of $A$. I'm having trouble formalizing this with ...
1
vote
1answer
34 views

Union operation on sets

I am reading an example in a book that says: The union of $ \{ x\in \Bbb R\; | \; x \lt 5\} $ and $ \{ x\in \Bbb Z\; | \; x \lt 8\} $ is $\{ x\in \Bbb R\; | \; x \le 5, or\;x = 6\; or\; x = 7\} $ I ...
0
votes
1answer
16 views

Meaning of notation $\{x,y\}\subset A$ in a partition

I recently came across this notation: $$\{x,y\}\subset A$$ Where $A\in \mathbb{A}$, and $\mathbb{A}$ is a partition of a non-empty set X. Does it mean that $x,y\in A$? Isn't $A$ a set of elements and ...
3
votes
1answer
31 views

Prove the relations of the cardinality of sets

Let us define natural numbers in the following manner (also assume Peano Axioms), $$\varnothing=0\\n^{+}=n\cup\{n\}$$where $n^{+}$ is the successor of $n$. Let $E$ and $F$ be two sets such ...
0
votes
1answer
18 views

Determination of an uncountable set

Which of the following sets are uncountable? $\{f:f:\mathbb N\to \{1,2\}\}$ $\{f:f:\{1,2\}\to \mathbb N\} $ $\{f:f:\mathbb N\to \{1,2\}, f(1)\leq f(2)\}$ $\{f:f:\{1,2\}\to \mathbb N, f(1)\leq f(2)\} ...
39
votes
9answers
3k 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 ...