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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2
votes
1answer
31 views

Show that $\left|X\right|=\left|Y\right|$

Let $A, B$, two sets. $X$ is the set of all relations from $A$ to $B$, and $Y$ is the set of all functions from $A$ to $P(B)$ (power-set of $B$). Prove that $\left|X\right|=\left|Y\right|$. My ...
0
votes
0answers
10 views

For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
0
votes
1answer
30 views

Prove the formula $ff^{-1}(B) = B \cap f(X) \subset B$ where $f: X\to Y$

Still struggling with proofs. This formula was presented as a given in my book and it wasn't intuitive to me at all, so I wanted to verify it as it seems images and inverse images play important roles ...
1
vote
2answers
79 views

What does $\vee$ mean in set theory?

The following proof is from Probability by Davar Khoshnevisan. There is a symbol $\vee$ in the third sentence of the proof. What does this symbol mean, please? There seems no definition about it in ...
0
votes
1answer
40 views

Elementary question on the intersection of sets.

I was curious about set intersections. I know that $\{x\}=\lim_{n\to\infty}[x-\frac{1}{n},x+\frac{1}{n}]\;\forall x\in{R}$. Then, what would be the followings? $(i) ...
1
vote
2answers
30 views

Order Type of $\mathbb Z_+ \times \{1,2\}$ and $\{1,2\} \times \mathbb Z_+$

I'm currently working on §10 of "Topology" by James R. Munkres. I've got a problem with task 3: Both $\{1,2\} \times \mathbb Z_+$ and $\mathbb Z_+ \times \{1,2\}$ are well-ordered in the ...
2
votes
0answers
56 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
2
votes
3answers
90 views

Notation for a space of finite sequences

For a given set $X$, what is the notation for the space of all finite $X$-valued sequences? I realise that the space of $n$-tuples is written as $X^n$, and the space of infinite sequences is ...
1
vote
0answers
17 views

Is there notation or a name for the complement of the unbounded face of a planar graph?

Let $G$ be a finite graph embedded in $\mathbb{C}$. Let $F$ denote denote its unbounded face. Is there notation or a name for $F^c$ without referring directly to $F$. Of course this is equivalent ...
0
votes
3answers
40 views

A non-principal ultra filter containing the even numbers, need hint now.

I posted a question about an exercise asking to prove that there exists a non-principal ultra filter on N containing the set of even numbers. My original post asked about a possible answer. It was ...
2
votes
1answer
20 views

Domain of a composite function

I was given the question: Find the domain of the function $f(x)=\ln(\ln(\ln x))$ I found the answer by inspection: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
3
votes
1answer
81 views

Is there an established notion for the square root of a set?

I'm looking for reading I could do around the concept of square rooting a set. I'm defining$\sqrt{A}$ to be the largest $B$ (by $\subseteq$), s.t. $B^2 \subseteq A$. So $\sqrt{A\times B} = A \cap ...
0
votes
0answers
37 views

Simple Math Problem on Interval

It's not clear for me. I see this wikipedia page for a difference of half interval on $\mathbb{R}$ and interval on $\mathbb{R}$? For example $$ \{ (-\infty \le x \le a) \, \left|\, a \in \mathbb{R} ...
1
vote
4answers
42 views

Proof simplification

I am tasked with proving the following: $$\varnothing - A = \varnothing $$ My Attempt : Assume there exist $x \in $$\varnothing - A $ then $$ x \in \varnothing - A \Rightarrow x \in ...
1
vote
1answer
20 views

Cardinality Borel $\sigma$-algebra

I'm reading a proof about the cardinality of the Borel $\sigma$-algebra and it uses some notions from set theory where I'm not all that familiar with. So the idea is that we have a set $U = ...
1
vote
0answers
21 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
0
votes
0answers
22 views

Is this set A equipollent to $\mathbb{R}$ or to $\mathbb{R}^\mathbb{R}$? [duplicate]

I have to find out whether the set A consisting of all bijective functions from $\mathbb{R}$ to $\mathbb{R}$ is equipollent to $\mathbb{R}$ or to $\mathbb{R}^\mathbb{R}$ and prove it. I think it is ...
1
vote
2answers
22 views

Size of a set of Well Orders

Given a set $X$ of size $\kappa$, is there any way to work out the number of well-orders on $X$? It seems like it should be possible but I can't see how to do it. Surely if $X$ is infinite then $X ...
0
votes
1answer
24 views

Understanding the definition of the d-dimensional Hyperube

Please see the picture bellow about the definition of the nodes of the d-dimensional Hypercube. Could anyone please tell me what does that notation means. I get confused with the superscript after the ...
0
votes
0answers
28 views

Equivalence relation in measure theory

Every set of positive measure has non measurable subsets and, moreover: $\mathcal{P}(A) \subseteq \mathcal{L} \iff \lambda(A)=0$ How would you go about proving this? Cant get my head around it ...
1
vote
3answers
30 views

Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
1
vote
1answer
16 views

Proving equality of functions using their restrictions

I have been going through Elementary Set Theory by Enderton and once again I am stuck on an exercise, which goes like this (p.88, exercise 27): Assume that $A$ is a set, $G$ is a function, and ...
0
votes
0answers
12 views

Show that the completion of a ring $\xi$ is a $\sigma$ - algebra

$\mathcal{N}$ be the class of $\mu$ - null sets. Let $\mathcal{S} \bigtriangleup \mathcal{N}:= \{ E \bigtriangleup N : E \in \mathcal{S}, N \in \mathcal{N} \}$. Similiarly $\mathcal{S} \cup ...
-1
votes
1answer
21 views

Countable cartesian products

What is a countable cartesian product of finite sets? That is, suppose $A_j$ are finite sets. Then what can you say about $\prod_{n=1}^{\infty} A_j$? I know countable cartesian product of countably ...
0
votes
2answers
62 views

Solve set problems without Venn diagrams

How to solve set problems without the aid of Venn diagrams? Example: In a school 70% of students like hamburgers, 60% like pizza. 50% of them like both, hamburger and pizza. How many percents ...
3
votes
1answer
81 views

Infinite Direct Sums Vs. Infinite Direct Products

Let $|R|=|S|=\infty$. In very many concrete categories, I know $R^S$ can be identified as the set of all functions from S to R, and the much "smaller" $R^{\oplus S}$ can be identified as the subset of ...
0
votes
0answers
15 views

Bijective Functions between Multiple Dimensions [duplicate]

Do bijective functions exist that map from a function of one dimension to a function of another dimension? For example, does there exist a function $f : \mathbb{R^2} \rightarrow \mathbb{R^3}$ that is ...
0
votes
0answers
20 views

Lebesgue measure and symmetric difference inequality

Suppose the conclusion holds. Therefore $\begin{align} \lambda(A \bigtriangleup B) < \epsilon &= \lambda(A \B \cup B\A) < \epsilon \\ &= \lambda(A \cup B \ - A \cap B) < \epsilon ...
1
vote
1answer
35 views

Calculate the number $o(\mathbb{R})$ of open subsets of the real line. [duplicate]

Calculate the number $o(\mathbb{R})$ of open subsets of the real line. I know that the answer is $\mathfrak{c}$ but I don't know how my lecturer got this. I am doing an introductory topology course, ...
1
vote
2answers
62 views

Why is this not a $\sigma$ - algebra

Why is $\displaystyle \{\cup_{i=1}^{\infty} (a_i, b_i] : a_i < b_i, i=1,...,n, n \in \mathbb{N_0} \}$ not a $\sigma$-algebra? It looks ok to me, the only way I can see that this would fail is ...
2
votes
2answers
47 views

Which of the following is NOT true? please see the options listed below.

a. The union of two countable sets is countable. b. A subset of a countable set is countable. c. If A is an uncountable set, and B is countable, then A - B is uncountable. d. The cardinality of ...
1
vote
2answers
42 views

Help Me Understand: Proof that Finite Intersection of Open Sets is Open

The proof is here: (link). I don't see how the third line (starting with Thus: $\exists \epsilon_i$...) is justified. That is: just because $x \in U_i$, for all $i$, how do I know that a ...
1
vote
2answers
77 views

A set of all the natural numbers can biject to a set of all the even numbers? Is this a disprove?

I saw that thing that if you biject set of all the natural numbers and a set of all the even numbers, they are able to pair. I don't think you can compare two infinites just like this I believe ...
0
votes
1answer
66 views

Extensional versus intensional theories in mathematics

Lambda calculus is often cited as an intensional theory whereas set theory is cited as an extensional theory. What are other examples of extensional and intensional theories of mathematical logic?
0
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2answers
28 views

Greatest lower and least upper bounds for a set of pairs

Had some trouble with this question in an exam recently, and wanted to make sure I reasoned correctly. The question was: $X$ is a set of pairs of real numbers $(x,y)$, with absolute values less than ...
0
votes
4answers
147 views

Singleton sets are a subset

I am tasked with prove the following elementary result. I am concerned about being rigours enough in my proof: $$a\in S \iff \{a\} \subseteq S $$ My Attempt: Suppose $\{a\} \subseteq S $ . Then ...
1
vote
1answer
45 views

show that for an infinite cardinal $k$, $k + k = k$

Show that for an infinite cardinal $k$, $k + k = k$ So far I have that $k + k = 2k$ Is it possible to somehow show that $2k = k$? I've been trying to understand some cardinal arithmetic, and I ...
1
vote
0answers
19 views

Is there a name for the corresponding notion of inductive subset in the context of well-ordered sets?

This is a question of terminology. I can't avoid being a little verbose before getting to it. The principle of mathematical induction states that, for any subset $e$ of $\omega$ (the set of natural ...
1
vote
1answer
37 views

A non-principal ultra filter containing the even numbers

I just started trying to read through Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, by Robert Goldblatt. I'm near the beginning in the section on filters. One of the exercises ...
5
votes
1answer
338 views

Term for: There Exists a Rational between every two Rationals?

The integers and the rationals have the same cardinality, but the rationals satisfy the property that: $$ \forall p,q\in\mathbb{Q},\quad \exists r\in\mathbb{Q}\quad \textrm{s.t.}\quad p<r<q, $$ ...
1
vote
2answers
66 views

Proof makes sense?

$\exists! {A} \subset {Z}$ such that $A \cup B = A$, where $B$ is any subset of $Z$. Proof: Assume two such sets exist, $A_1$ and $A_2$ If $A_1 \cup B = A_1, \forall B \cup Z$, then $A_1 = Z$ If ...
3
votes
1answer
23 views

Language clarification in an article about filters

I started reading these notes. After enumerating four properties of a filter $\mathcal F$ in a topological space $(X,\tau)$ (1) $X\in\mathcal F$; (2) $V\in\mathcal F\wedge V\subseteq ...
1
vote
3answers
140 views

What is the meaning of the symbol \stackrel {<}{\neq}?

In the book Good Math by Mark C Chu-Carrol there is the following formula on page 130: $$ x ∈ (A ∪ B) \stackrel{<}{\neq} x∈ A ∨ x ∈ B $$ I don't know the name or meaning of this symbol and I can't ...
3
votes
3answers
74 views

How to prove $ A \cup \{a\} \approx B \cup \{ b \} \Rightarrow A \approx B $

How to prove this without recurring to cardinality? $ A \cup \{a\} \approx B \cup \{ b \} \Rightarrow A \approx B $ Where by "$ \approx $" I mean that there exists a bijective function between A and ...
2
votes
1answer
62 views

Behaviour of sum of $2^\kappa$ for all $\kappa<\lambda$ when $\lambda$ is singular [closed]

What can we say about the conditions under which $\sum\limits_{\kappa<\lambda}2^\kappa \leq \lambda$ holds when $\lambda$ is singular?
0
votes
1answer
31 views

Showing a set is well-ordered

Let $(C,S)$ be a well-ordered set. Let $d \notin C$. We define the set $D=C \cup \{d\}$ and the relation $S'=S\cup (C \times \{d\})$. Show the set $(D,S')$ is well-ordered. Any help would be much ...
0
votes
1answer
25 views

How to describe sets derived from same superset in ordinary language?

How do we describe two related sets in comparative terms? For example, S1({{x}, {x}} : x ∈ N) and S2({{x}, {x}} : x ∈ N : 0 less than x less than 5)? Do we say that we have imposed a set rule (or set ...
0
votes
1answer
21 views

How to describe set expressions in ordinary language?

What is the proper way to describe a set in ordinary language? For example, say we have the expression: S1({{x}, {y}} : x ∈ A : y ∈ B). I assume in this case we just say that S1 is a set of ordered ...
0
votes
2answers
46 views

Find the $A \cap B$ for given sets

Let $A$ and $B$ be two non empty and defined sets $$A = \{(x,y) : y = e^x , x \in \mathbb{R} \},$$ $$B = \{(x,y) : y = x , x \in \mathbb{R} \}.$$ What is $A \cap B$ ? How to write Set A and B in ...
2
votes
1answer
62 views

How far is it possible to develop cardinals without ordinals?

I'm wondering which of the usual facts about cardinals in ZFC can be established without using ordinal arithmetic at all. After all the definitions of a cardinal (as a class of equivalence), and also ...