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
1answer
26 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
-2
votes
2answers
55 views

Proving that $A\cup\emptyset=A$ and $A\cap\emptyset=\emptyset$

I need to prove that $A\cup\emptyset=A$, and $A\cap\emptyset=\emptyset$. It's seem like it's obvious, yet how can I prove it mathematically?
1
vote
2answers
82 views

Is the set of all pairs of real numbers uncountable?

My hypothesis is that $\mathbb{R \times R}$, the set of all pairs $(r_1, r_2)$, of real numbers is uncountable. I understand that the set of all pairs of natural numbers is countable. But could ...
0
votes
1answer
23 views

Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...
0
votes
1answer
38 views

Can an element of a power set $2^A$ be a subset of $2^A$?

This question is continued from a previous thread I started, but it had more than one question so I had to move the other question here. For this example consider an injective map $f: A \to 2^A$ then ...
0
votes
1answer
50 views

If $A$ is a non-empty set and $2^A$ is the power set of $A$. Is $2^A \subseteq A$?

I'm aware that if there exists an injective map $f: A \to 2^A$ then for each element $a\in A$ $\exists$ $f(a)\subseteq A$. But does this also mean $f(a)\subseteq 2^A$? I ask this because when ...
6
votes
6answers
160 views

Is the set of all pairs of natural numbers countable? [duplicate]

Say that $\Bbb N \times \Bbb N$ is the set of all pairs $(n_1, n_2)$ of natural numbers. Is it countable? My hypothesis is yes it is countable because sets are countable. But I am unable to come up ...
0
votes
2answers
43 views

Closure of $\Bbb R$ [on hold]

Is the closure of $\Bbb R$ equal to $\Bbb R$ itself or the extended real numbers $\bar{\Bbb R}$? Thanks for any comment.
5
votes
1answer
91 views

Is this enough to explain why set theory work in real analysis?

Sorry for starting a lot of topics in set theory; I think this will be my last. I just want to know what are the basics I need to know about set theory to mathematical analysis. Is what I have ...
0
votes
5answers
77 views

A better proof for the set of irrational number not closed under ordinary multiplication.

A positive irrational number $$q$$ is by definition a real number than cannot be expressed as a ratio of $2$ integers. To show that the set of irrational number is not closed under ordinary ...
0
votes
1answer
29 views

Show that the set $\mathbb{Q}^+$ is a group under ordinary multiplication

To be a group, a set with a binary operation has to satisfy all four of the group axioms. My problem is with closure as each time I am unsure if my proof suffices. The set of positive rational numbers ...
1
vote
2answers
52 views

Is the fact that these sets can not exist a consequence of Russels paradox?

Some time ago I asked why a given collection of objects could not be a set(something to do with abstract algebra). I got three answers, one was close to the Russel paradox, two other explanations ...
2
votes
0answers
34 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
1
vote
1answer
12 views

Is it overkill to define the closure of a set $A,A\subseteq B$ by the union of the range of the recursive function $h(0)=A, h(n^+) = h(n)\cup f[h(n)]$

$f:B\to B,A\subseteq B$. Is it overkill to define the closure of a set $A,A\subseteq B$ over $f$ by the union of the range of the recursive function $h(0)=A, h(n^+) = h(n)\cup f[h(n)]$? I ...
3
votes
3answers
88 views

If $A$ and $B$ are sets, then either $A \in B$ or $A\notin B$

Given that $A$ and $B$ are two sets, is the following proposition a tautology: $A\in B \vee A\notin B$. I do not know any set theory beyond the naive one.
0
votes
2answers
49 views

What kind of set-theory is sufficient to understand mathematical analysis?(book recommendation))

I am looking for books with set theory and logic that is sufficient to understand mathematical analysis. I guess another question might be if there even exists such a book. There are basically two ...
0
votes
2answers
39 views

$A \subset B \implies f^{-1}(A) \subset f^{-1}(B)$

Prove: $A \subset B \implies f^{-1}(A) \subset f^{-1}(B)$ I am busy setting up a proof for Real Analysis, and have come to a point where I need to use the above statement. Intuitively, I ...
0
votes
0answers
30 views

Which of the following sets have the cardinality the same as $R$ [duplicate]

Which of the following sets has the same cardinality as that of $\mathbb{R}$? $W$=The set of constant functions on $\mathbb{R}$ $X$=The set of polynomial functions on $\mathbb{R}$ $Y$=The set of ...
0
votes
1answer
60 views

Proving set identities

I am attempting to work on some proofs for my math assignment, but I'll be honest in that I am really struggling to understand them. I read through the power point given by my teacher; however, even ...
0
votes
1answer
26 views

simplifying set theory expression

I'm trying to simplify a set theory expression, it relates to a programming problem... I have a set of staff members who are grouped into various teams (Team A, Team B, Team C etc ...up to Team H). ...
1
vote
2answers
22 views

Set Theory - Simplify expression

Can the following be simplified? It's been a long time since I did set theory and I don't remember my simplification rules. This is probably totally easy... can I simplify this any further? $(A \cap ...
0
votes
0answers
27 views

Cardinality and Set Theory

For cardinals k which is cardinal of K and l which is cardinal of L and m which is cardinal of M. W.T.S [ $ k^{m+l} $ = $ k^m k^l $] by constructing a bijective function F : $ ^MK \times ^LK \to ...
-4
votes
0answers
21 views

If the function f and function g is one to one, show that the composition g of f is one to one? [on hold]

If the function $f$ and function $g$ is one to one, show that the composition $g$ of $f$ is one to one? How do I show that the composition of two function both of which are one to one produces a ...
1
vote
2answers
46 views

Show that $f: \mathbb N \to \mathbb N$, $f(x)=x^2$ is not onto

To begin, the definition of an onto (surjective) function is as follows. A function $\phi$ from $A$ to $B$ is surjective if for each for each $b$ in $B$, there exists at least one $a$ in $A$ such ...
-4
votes
1answer
92 views

“Notes on Set Theory” - Badly Written? [on hold]

I hope I'm not breaking any rules by asking this particular question, but I honestly can't think of a better place to inquire about this. A few days ago I managed to get my hands on "Notes on Set ...
0
votes
1answer
22 views

Are maximal intervals of open nonempty sets always equal?

Let $O\subset\mathbb{R}$ be an open nonempty interval. Define for every $x\in O$: $$a_x = \inf\{a\in\mathbb{R}\mid(a,x]\subset O\}$$ $$b_x = \sup\{b\in\mathbb{R}\mid[x,b)\subset O\}$$ $$I_x = (a_x, ...
4
votes
1answer
48 views

Product of two sets with density zero has density zero?

Let $A$ and $B$ be two subsets of $\mathbb N$ which have asymptotic density zero. Define $A\times B$ as the set of integers of the form $ab$ with $a\in A$ and $b\in B$. Must $A \times B$ also have ...
1
vote
2answers
58 views

Given nonempty sets S and T, does there exist a set R that is disjoint from S with |R|=|T|?

Let $S$ and $T$ be nonempty sets. I would like to show that there exists a set $R$ such that $S\cap R=\emptyset$ and $\left\vert T\right\vert = \left\vert R\right\vert$. Here is my work so far. Let ...
0
votes
0answers
13 views

Set-sum in $\mathbb Z_p^d$

Notation: $G$ abelian group, $X,Y \subset G$, $X^c = G \backslash X$, $|X| =$ number of elements of $X$, $X+Y = \{ x+y \in G; x \in X, y \in Y \}$. Let $B \subset G$ fixed and $\lambda(x) = ...
1
vote
1answer
20 views

Set builder notation for matching element pairs

I have a set of pairs, $S = \{ \langle a,b \rangle_1, \langle a,b \rangle_2, ..., \langle a,b \rangle_n \} $ where $a$ is not unique amongst the pairs. If I want to express the extraction of all the ...
1
vote
3answers
58 views

Which of the following statements are false?

All of the following sets are subsets of positive integers. $A = \{x\mid x\ \text{is divisible by 2}\} \\ B = \{x\mid x\ \text{is divisible by 4}\} \\ C = \{x\mid x\ \text{is divisible by 6}\}$ ...
0
votes
1answer
25 views

A question about the well-ordering theorem

In Munkres' book "Topology", he states that Well-Ordering Theorem: If $A$ is a set, there exists an order relation on $A$ that is a well-ordering. Then, in his discussion of the Maximum ...
0
votes
2answers
22 views

How to proof equivalence relation?

I need help with this problem: Let $S=\left\{\left[\begin{matrix} a & b \\ c & d \end{matrix}\right] : a,b,c,d \in \mathbb{C}\right \}$ and $M=\left\{\left[\begin{matrix} a & b \\ ...
1
vote
2answers
35 views

Concept of an Equivalence class partition

The definition: Part 1: The equivalence classes of an equivalence relation on a set S constitute a partition of S. Part2: Conversely, for any partition of S, there is an equivalence relation on S ...
1
vote
1answer
82 views

A challenge in Prof.Terence Tao's book “Analysis”: Using axiom of specification to define image of a function

On page 64 (3.4 Images and Inverse Images) of "Analysis I" by Terence Tao, it says: Note that the set $f(S)$ ($f$ is a function) is well-defined thanks to the axiom of replacement (Axiom 3.6). ...
5
votes
1answer
33 views

Sets raised to exponents

"Find two non-empty sets $A$ and $B$ for which $A^B$ and $B^A$ are not the same size." I'm really not sure what this means or how to even go about attempting this... Can anyone provide an example of ...
3
votes
2answers
43 views

What does the multiply sign mean in set

A question about set notation. What does the multiply sign here mean? $$\omega = \times_{i\in N}T_{i}$$
3
votes
4answers
63 views

Can a nonempty set ever equal its Cartesian product with another set?

Suppose that $S$ and $T$ are sets, with $S\neq \emptyset$. Would it be possible to have $S=S\times T = \{(s,t): s\in S, t\in T\}$? If such were the case, then we'd have \begin{align*} \{(s,t): s\in S, ...
2
votes
1answer
12 views

Show that a particular set is not a limit ordinal (aim is to define ordinal subtraction)

Let $\alpha$ and $\beta$ be two ordinals with $\beta \leq \alpha$. Define $$ X:= \{\gamma \in \alpha^{+} : \beta + \gamma \leq \alpha\}.$$ I have shown this is an ordinal. Now I need to show it ...
2
votes
1answer
25 views

Choice function for collection of arbitrary finite sets. AC required?

I understand how we can show the existence of a choice function for any (finite or infinite) collection of (finite or infinite) subsets of, say, $\mathbb{N}$ or $\mathbb{Z}$ without using the axiom of ...
0
votes
1answer
46 views

Are there ordinals beyond all the $\omega$'s? [on hold]

Are there ordinals that are somehow "beyond" all the $\omega$'s?
0
votes
1answer
25 views

Show that the function $g$ is bijective

Let $~f\colon X \!\to\! \{i \in \mathbb{N}:\! 1 \leq i \leq n \}$ be a bijective function and $~x$ be an element of $~X$. Now define the function $g\colon X-\{x\}\!\to\!\{i \in \mathbb{N}:\! 1 \leq i ...
1
vote
2answers
67 views

How to prove that $C\cdot\aleph_0=C$

How can I prove that $C\cdot\aleph_0=C$? I tried this: Given that $k\cdot 1=k$ and $C\cdot C=C$ if $C\cdot C = C \wedge C\cdot 1 = C \wedge C>|\mathbb N|>1$ then $C\cdot |\mathbb N|= C$ c is ...
5
votes
2answers
95 views

Is the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?

I know the closure of $\mathbb Q$ is $\mathbb R$, but does this imply that the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?
0
votes
0answers
30 views

Finding the compliment of a logical expression in respect to another logical expression.

What I would like to do is to find the logical compliment of one expression in respect to another logical expression. If possible, I would like to know if there has been work in this area - I haven't ...
-1
votes
1answer
51 views

Binary strings and discrete math

Question: Let $S$ be the set of binary strings of length $30$ with $10$ $1$’s and $20$ $0$’s. Let $A$ be the set of the first $30$ positive integers $\{1,2,3,\dots,30\}$. Let $B$ be the set of all ...
2
votes
0answers
25 views

Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
2
votes
2answers
32 views

Inverse mapping on a set $U_1\times U_2$, wrong intuition?

Let $f(x) = (f_1(x),f_2(x))$ where $f: X\to Y_1\times Y_2$. And $f_1:X\to Y_1, f_2: X\to Y_2$ where $X,Y_1,Y_2$ are topological spaces. I want to prove some continuity properties, but my ...
0
votes
1answer
53 views

Need help with set theory questions

So I was in the hospital for two weeks, and missed quite a bit of new material. Unfortunately, the text book I've got and my friend's notes aren't very helpful in catching up with the missed material, ...
0
votes
0answers
23 views

K-theory,proper class,set,isomorphism types

Define $K$ as the free abelian group with generators $[A],[A'],[A'']$,the equivalence classes of isomorphism types, modulo $[A]=[A']+[A'']$ where $0\to A'\to A \to A''\to0$ is a s.e.s. of modules in ...