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, (un)...

learn more… | top users | synonyms

0
votes
2answers
36 views

The cardinality of all the infinite binary sequences that don't contain 010

Find the cardinality of all the infinite binary sequences that don't contain 010 I think it's $\aleph_0$. I marked the set all infinite binary sequences that don't contain 010 in A, and the set of ...
2
votes
1answer
35 views

Well-ordering principle and theorem

Could somebody clearly explain the difference between the well-ordering principle and the well-ordering theorem? Is one of these related to the Principle of Mathematical Induction, and the other to ...
1
vote
1answer
53 views

Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
1
vote
1answer
32 views

Reasoning informally about $\{x \in B \mid x \notin C\} \in \mathscr P(A)$

Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
3
votes
1answer
20 views

Prove that $\text{Dom } (S\circ R) ⊆ \text{Dom }R $

Let $R$ be a relation from $A$ to $B$ and $S$ be a relation from $B$ to $C$. Suppose, $x \in \text{Dom }(S\circ R)$. Then, it follows that there $\exists y \in C$ such that $(x,y) \in S\circ R $. ...
3
votes
2answers
23 views

Rewriting $\mathscr P(\bigcup_{i \in I} A_i)\not\subset\bigcup_{i \in I} \mathscr P(A_i)$ in more fundamental terms.

Working through Velleman's "How to Prove It" and currently having a bit of difficulty with a problem where I'm asked to rewrite this: $$\mathscr P\left(\bigcup_{i\in I} A_i\right)\not\subset\bigcup_{...
3
votes
1answer
29 views

Commonplace sets

I recently started reading about sets of numbers, set builder notation, and operations on sets of numbers. To practice using different symbols (e.g., $\wedge$) and different set "layouts," I decided ...
2
votes
3answers
44 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
1
vote
2answers
33 views

Cardinality of subsets with finite intersections

Let $\ F_0 $ be a family of disjoint subsets of $ C$. $\ |C|= \aleph_0$. Prove that $\ (*) |F_0|\leq\aleph_0 $. This part was relatively simple, in the presence of choice an injection can be ...
0
votes
0answers
15 views

Chartrand Mathematical Proofs 3e Exercise 1.45

I'm self-studying this book to learn how to do proofs, I have previously studied Calculus 1,2,3 and Linear Algebra in college in the US. I have a problem with the following question: Exercise. 1.45 ...
2
votes
1answer
34 views

Existence of an inverse relation for $R \subseteq A \times A$.

I'm stuck with the following problem: Given the set $A = \{1,2,3,4,5\}$, construct a relation $R \subseteq A \times A$ such $$ R \circ R^{-1} = \triangle_A = \{(a,a) \hspace{5pt} | \hspace{5pt} a \...
1
vote
1answer
28 views

Proving that $A+B - (A \cap B) = A \cup B$ for binary integers

I hope computing questions are fine here. I'm trying to show that for all binary numbers $A$ and $B$, $A+B - (A \cap B) = A \cup B$. It's confusing me firstly because I'm not sure what the "set ...
1
vote
0answers
33 views

For each of the following sets, determine its cardinality (ω, 2ω, or something else) and prove that your answer is correct

(a) A1 = {f ∈ (ω → ω) : ∀n,m ∈ ω (n < m ⇒ f(n) < f(m))}. (b) A2 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω f(m) ≤ n}. (c) A3 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω (n ≤ m ⇒ f(n) = f(m))}. a) A1 = {f ∈ (ω → ω) : ∀n,...
4
votes
0answers
86 views

Every family $\mathscr{A} $ of sets satisfies $|\mathscr{A} \setminus \mathscr{A}| \geq |\mathscr{A}|$

Let $\mathscr{A} $ be a set of sets. Let's denote $\{A \setminus B : A,B \in \mathscr{A}\}$ by $\mathscr{A} \setminus \mathscr{A} $. The Marica-Schönheim theorem in combinatorics says that $|\...
0
votes
1answer
38 views

Assistance with finding the accumulation points for $(3,6) \cup (6,9]$

I'm having trouble digesting the definition of an accumulation point(s). Can you help me to understand it given the following: $(3,6) \cup (6,9]$ I know this produces the interior set $(3;9]\...
1
vote
1answer
53 views

Proving the Cardinality of a set in R

Let $\ A\subset R $ have the following characteristic: For all $\ a,b \in A$ , $\ \frac{a+b}{2} \notin A$. Prove that there exists a maximal set A. Prove its cardinality is $\ \aleph $. The first ...
2
votes
2answers
54 views

A 'bad' definition for the cardinality of a set

My set theory notes state that the following is a 'bad' definition for the cardinality of a set $x:$ $|x|=\{y:y\approx x\}$ $(y\approx x\ \text{iff} \ \exists\ \text{a bijection}\ f:x\rightarrow y )...
2
votes
2answers
51 views

Finding counterexamples in elementary set theory.

I had the following two problems: Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for ...
2
votes
2answers
38 views

About the minimal equivalence relation identifying some points.

I am solving a problem where I have a set $X$ together with a subset of elements that I want to identify. To do this I consider the minimal equivalence relation identifying these points. I have a ...
0
votes
1answer
36 views

Prove any function can be written as a composition between an injective and a surjective function.

Given an arbitrary function $f:A\rightarrow B$, write it as a composition between an injective and a surjective function, respectively.
-1
votes
1answer
28 views

What is the correct set elements for these

I have the following specifications: $U = \{x: x$ is an integer, $0 < x < 10\}$ $M = \{x: Y – 1, Y$ is prime number less than $10\}$ $N = \{x: x^2< Z, Z$ is the smallest prime number$\}$ ...
0
votes
1answer
46 views

Why is it that if $A \subseteq B$ then $\overline{A} \subseteq B$?

In a solution that I was reading, we were required to prove that $$\overline{A} \subseteq f^{-1} (f(A)) \tag{$*$}$$ The author did (roughly speaking) the following. First prove that $A \subseteq f^{...
0
votes
1answer
13 views

If unions of two families sets are disjoint then families of sets are disjoint too.

I have read that theorem "Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. If $\cup\mathcal{F}$ and $\cup\mathcal{G}$ are disjoint, the so are $\mathcal{F}$ and $\mathcal{G}$" is ...
0
votes
1answer
41 views

Existence and Uniqueness of an ordinal

$\underline{\mathbf{Problem:}}$ If $\alpha < \beta $ then $\exists$ a unique $\gamma$ st. $\alpha+\gamma=\beta$, where '$+$' denotes ordinal addition. If someone would be so kind to check the ...
0
votes
0answers
28 views

Multi correct objective question. [duplicate]

Let $A$ be any set. let $\mathbb{P}(A)$ be the power set of $A.$ Then which of the following is/are true about $\mathbb{P}(A).$ $1.~ \mathbb{P}(A)=\emptyset$ for some $A.$ $2.~\mathbb{P}(A)$ is ...
1
vote
1answer
64 views

The union of $(3,6) \cup (6,9]$

$(3,6) \cup (6,9]$ I got $[4,5,7,8,9]$, which is missing an element in the interval. I'm supposed to take the union and find the interior, boundary points, and accumulation points, but not sure how ...
1
vote
0answers
20 views

Recurrence relation has bijection

I have a question regarding one-to-one mappings. I have two sets A = {1,2,3,4,5..} and B = {1,4,7,10,13,...} I have to create a bijection between the maps. The easiest one if f(n) = 3(n)-2 However, ...
0
votes
0answers
74 views

Which sets are finite, countable, countably infinite, and uncountable?

I believe I have these all correct, but if I made an error could you lend a hand and possibly explain why I was mistaken? Thanks in advance. Consider the following sets: $X_{1}=\emptyset$ ...
0
votes
1answer
34 views

Finding the Union and Intersection of the indexed collection?

For each natural number $n≥3$ let, $A_n$ = $[\frac1n, 2+ \frac1n]$ and $\mathscr A\ =${$A_n :$$n≥3$} So what I did first to solve this problem is I plug in natural numbers greater than or equal to 3....
2
votes
4answers
77 views

Proof that $A \cap B$ and $A \setminus B$ are disjoint.

I am trying to prove that $A \cap B$ and $A \setminus B$ are disjoint. Here is what I've done so far. Is there anything that's wrong in my proof, and is there anything that can make it better? ...
1
vote
1answer
65 views

What is meant by $sup(A\cup B )$

I am given two subsets $A,B$ of $\mathbb{R}$ which are not empty and are bounded above. Now according to a lemma, both of these sets have supremums. My issue is part of the question deals with sup$(...
2
votes
2answers
86 views

Does there exist a surjective function from $(0,1)$ to $[0,1]$?

Can I say that cardinality of $(0,1)$ is less than the cardinality $[0,1]$ ?
0
votes
1answer
60 views

Nonempty closed sets on a connected space imply nonemptiness of intersection?

I am dealing with just real line to make things little easier for me. Suppose we have a set $X=[0,x],X'=[x,\infty)$. For the sake of argument, assume both are closed and nonempty. Claim: By the ...
1
vote
3answers
49 views

A well-order on a uncountable set

I can't find an example of a well-order on an uncountable set. Is possible to prove that exists with the Axiom of Choice? How can I give a pratical construction? I try to define a well-order on ...
2
votes
1answer
43 views

Properties of the power set of $A$

Let $A$ be any set . Let $\wp(A)$ be the power set of $A$. Then which of the following are true 1) $\wp(A) = \emptyset$ for some $A$ 2) $\wp(A) $ is a finite set for some $A$ 3) $\wp(A)$ is a ...
18
votes
6answers
4k views

How to represent “not an empty set”?

I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians? My idea is: $|A| > 0$ However, using the emptyset $\emptyset$ ...
1
vote
2answers
61 views

Explain Example on Maximal Element with sets

I am trying to understand maximal element and I cannot understand this example from Wikipedia As an example, in the collection $$S = \{\{d, o\}, \{d, o, g\}, \{g, o, a, d\}, \{o, a, f\}\}$$ ...
0
votes
2answers
30 views

Proving 1-1 correspondance

This question is giving me headaches some time now. I tried to solve it using compositions of functions, but with no use. Any help would be appretiated. If $f:{A}\rightarrow{B}$,$g:{B}\rightarrow{C}$,$...
9
votes
5answers
249 views

How to determine the existence of all subsets of a set?

Given The definition of subset; The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$ we know what a subset is and what a power set ...
-1
votes
0answers
23 views

Make the sum of array as zero

I have an array A[N] where the elements can be wither positive or negative but not zero Now I want to make the sum of all elements of array as zero. What is the minimum number of ways to do that. ...
0
votes
2answers
34 views

$x \in y \Rightarrow \mathcal{P}(x) \in \mathcal{P}(y)$?

my doubt is if $$x \in y \Rightarrow \mathcal{P}(x) \in \mathcal{P}(y)$$ is true, where $\mathcal{P}(A)$ is the power set of A: $\mathcal{P}(A)= \{T| T \subseteq A\}$.
1
vote
1answer
39 views

Diophantine relations using an equation with polynomials of degree at most 4

I'm completely stuck at exercise 5.8.5 of Mathematical Logic, Chiswell & Hodges: Here are the mentioned definition and theorem: I'm stuck because I failed to use the hint given in the ...
1
vote
2answers
14 views

Section and segment of a relation $R$

$\mathbf{Definitions:}$ $Z$ is a $R$-section of a set $X$ iff $Z\subseteq X$ and $x\in Z$ whenever $x\in X \ \land y\in Z \land \ xRy$, $\ $for some relation $R$ on $X$ The $R$-segment of a set $X$...
0
votes
1answer
32 views

How can I prove that $(X \times Y) \cap (Z \times T) = (X \cap Z)\times(Y \cap T)$

I have this problem that I'm trying to figure for more than an hour and I just can't even start it. Is someone able to help me? Any help is much appreciated!! PROBLEM: How can I prove that $(X \...
0
votes
1answer
35 views

Partition of unity from RCA Rudin

Let me ask the following question: How Rudin applies Theorem 2.7 in the begining? He take some $x\in K$ then $x\in V_i$ where $i=i(x)$. What's next? I thought on this about couple hours but no ...
0
votes
2answers
42 views

What is the minimum and maximum of a set with only one element?

This is surely a trivial question but I want to be sure I understand correctly what happens. Given a set $A = \{1\}$, what is $\min A$ and $\max A$? Is it $\min A = 1$ and $\max A = 1$?
0
votes
1answer
33 views

How to count the set $\mathbb{Q}$?

I've seen this http://www.homeschoolmath.net/teaching/rational-numbers-countable.php which only counts the positive numbers, but how can also count (or order) the non positive ones?
1
vote
1answer
42 views

How many employees are there in the office

In an office, $37$ people like $X$ drink, $32$ people like $Y$ drink and $41$ people like $Z$ drink. Also, $5$ employees like all the types of drinks and $15$ people like at least two types of drinks. ...
2
votes
1answer
36 views

Classification of an open set in real

Prove that open set in real line can be represented as ar most countable disjoint union of open intervals. I know that this question repeated many times in MSE but let me ask the following question. ...
1
vote
1answer
75 views

Is it The Axiom of Power Set that guarantees the existence of (all) subsets?

The Axiom of Power Set asserts that: For any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$. That is, if something is a subset of $S$, then it's a member ...