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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5answers
82 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
55 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
39 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
14 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
91 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.
1
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2answers
59 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
40 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
61 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
23 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 ...
1
vote
1answer
51 views

Proving $k^{m+l} = k^m k^l$ by constructing a bijective function F : $ ^MK \times ^LK \to ^{L\bigcup M}K $

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? [closed]

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 ...
0
votes
1answer
23 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
52 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
60 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
22 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
77 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
23 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
37 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
88 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
46 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
66 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
27 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
47 views

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

Are there ordinals that are somehow "beyond" all the $\omega$'s?
0
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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
71 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
97 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
31 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
26 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
56 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
24 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 ...
2
votes
3answers
58 views

Show that $\mathbb{R}$ is a disjoint union of $\mathfrak{c}$ sets of cardinal $\mathfrak{c}$

Show that $\mathbb{R}$ is a disjoint union of $\mathfrak{c}$ sets of cardinal $\mathfrak{c}$, where $\mathfrak{c} = | \mathbb{R} | = 2^{\aleph_0}$. I find this problem very interesting and very ...
0
votes
0answers
17 views

Whether a given collection is a set or not [duplicate]

I originally knew that a set is a concept that has no definition. However, today, in abstract algebra class, the professor told us that the collection of all fields E such that E/F is an alegbraic ...
0
votes
2answers
43 views

How do I find the equation of given Venn diagram?

How do I get the equation of given Venn diagram? What kind of steps do I want to follow ? Example Equation: \begin{equation*} A \cap B' \cup C \end{equation*}
1
vote
3answers
43 views

Find the symmetric difference of two finite sets

Find the symmetric difference, $A \triangle B$, where $A=\{6,7,8,9,10\}$, and $B$ is the set of odd integers between $5$ and $10$. The end points are exclusive. I think the only answer it could be is ...
1
vote
2answers
35 views

Understanding line of given proof

I have to understand a set of proofs and I don't understand the reasoning behind this line "This is an injection, if $g(b_1) = g(b_2)$ then $F_{b_1}$ And $F_{b_2}$ intersect, which we shown never ...
0
votes
1answer
23 views

Determining if the relation is a poset

We have this question on our final review sheet and I want to make sure I fully understand it before the exam. I looked up several examples but they did not help. Determine whether the following ...
0
votes
1answer
30 views

Ordinal Exponentiation problem

I've found a problem in Set theory that I can't really get my head around. The problem is: Under ordinal exponentiation find an ordinal $\mu$ such that $\omega < \mu$ and $2^{\mu } = \mu$ (where ...
3
votes
1answer
54 views

Why are power sets called power sets?

Why are power sets called power sets? What is so powerful about them?
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votes
2answers
27 views

The meaning of singleton and existence of {ø,{ø}}

I can't understand difference between ø and {ø}. What is difference between x and set that has only x as element? I think the concept of set is meaningful when it has more than 1 element. If "ø and ...
-1
votes
0answers
13 views

Show $|X|\leq |Y|$ if and only if there is a surjection $Y\rightarrow X$ [duplicate]

Let $X$ and $Y$ be sets. Show that $|X| \leq |Y |$ if and only if there is a surjection from $Y$ onto $X$.