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, ...

learn more… | top users | synonyms

1
vote
0answers
28 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
0
votes
0answers
12 views

tuple of tuples notation

Is the following notation right for indicating a $\mathit{m}-$tuple of $\mathit{n_{j}}-$tuples (I mean that each tuple of the $\mathit{m}-$tuple has a different number of elements)? ...
0
votes
0answers
30 views

If A and B are disjoint finite sets, use induction on $|B|$ to show that $|A \cup B|$ is finite.

I'm asked "If A and B are disjoint finite sets, use induction on $|B|$ to show that $|A \cup B|$ is finite. " The main reason I'm having difficulty with this is that it seems so obvious I don't know ...
2
votes
1answer
39 views

Question about the cardinality of sets and infinity

Let's say we have $\mathbb{N}$, the set of natural numbers: $\{1, 2, 3, 4, 5...\}$ ...which has a cardinality of infinity, and the set $A_x$ which consists of the variable "$x$" (so $\{x\}$). If I ...
1
vote
1answer
21 views

The range of a function $f : n^+ \to \omega$ has a largest element.

I am completely lost on this question: Assume that $n \in \omega$ and $f: n^{+} \rightarrow \omega$. Show that $ran\ f$ has a largest element. What I don't understand is that we are given no ...
1
vote
3answers
25 views

Why is this relation transitive?

On $A = \{1,2,3,4,5\}$, define the relation $R = \{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4),(5,5)\}$. According to my book this relation is transitive, but according this definition there has ...
0
votes
0answers
29 views

Mysterious membership-relation question in Enderton's `Elements of Set-Theory'

The question I am currently working on states (mysteriously): ``Simplify $\in^{-1}_{\omega} [\{7,8\}]$". For those who have the text, this is exercise #18 in Enderton's `Elements of Set Theory' on ...
1
vote
2answers
27 views

Subtracting a set from a set of set

Let say that I have a set of set X = { {a,b}, {a,c}} and I want to remove the element {a,b} from X. What is the proper way to write this subtraction operation. X \ {a,b} or X \ {{a,b}} I think ...
-1
votes
0answers
34 views

Cardinality of the set of complex numbers [duplicate]

Given the continuum hypothesis, does the cardinality of $\mathbb{R}$ ($\aleph_1$) equal the cardinality of the set of complex numbers? If not, what is the cardinality of $\mathbb{C}$? Would it be ...
0
votes
0answers
37 views

Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
0
votes
1answer
28 views

Proving theorems on relations

I came across the following three statements about relations. I understand why the statements are true, but I am not sure how to demonstrate them mathematically. In the following, $A,B$ are sets and ...
1
vote
2answers
96 views

Fun quiz: where did the infinitely many candies come from?

Story 1: Let there be a bowl $A$ with countably infinite many of candies indexed by $\mathbb{N}$. Let bowl $B$ be empty. After $1/2$ unit of time, we take candy number 1 and 2 from $A$ and put ...
0
votes
3answers
44 views

Show that $A \cup B = (A$ \ $B ) \cup (A \cap B) \cup (B$ \ $A)$

Let $A, B$ be finite sets. Show that $A \cup B = (A$ \ $B ) \cup (A \cap B) \cup (B$ \ $A)$. Deduce that $|A| + |B| = |A \cup B| + |A \cap B|$ These are obvious when considering Venn diagrams but ...
1
vote
1answer
24 views

Question about the proof of this lemma: If $\alpha$, $\beta$ are ordinals, then either $\alpha \subset \beta$ or $\beta \subset \alpha.$

Proof: Clearly $\alpha \cap \beta$ is an ordinal, $\alpha \cap \beta = \gamma.$ Then $\gamma = \alpha$ or $\gamma = \beta$. For, if not, then $\gamma \not= \alpha$ and $\gamma \not= \beta$. Then ...
1
vote
1answer
36 views

Halmos Set theory - what is meant by this set (power sets chp 5)?

What does set D mean here? Could someone please explain in words what it is a set of? How does the sentence follow from it and can someone please translate the sentence? And how do De Morgan's laws ...
1
vote
2answers
44 views

“Unclosure” on a set with binary operation

I was wondering if there is any usefulness to having a set that has no closure under a particular operation. For example, the set of prime numbers, $\mathbb{P}$ along with multiplication of integers ...
0
votes
0answers
42 views

How to convince a layman that there are as many rationals as integers [duplicate]

I am having this problem with a friend of mine(he's an English major). I am well aware of the reasoning to show that there are as many rationals integers(By showing a one to one corresponding between ...
1
vote
0answers
32 views

Which basis orders [for the natural numbers] have been proven?

The set $A$ of nonnegative integers is called an additive basis of order $h$ if every nonnegative integer can be written as the sum of $h$ not necessarily distinct elements of $A$. For example, the ...
0
votes
5answers
57 views

Is it the case that for all sets $A, B, C,$ and $D$, $(A \times B) \cup (C \times D) = (A \cup B) \times (C \cup D)$?

I think I have managed to work this out, however there is no solution provided with the question so I thought it best to check. My working: Let $(x,y) \in (A \times B) \cup (C \times D)$ ...
-2
votes
1answer
18 views

Question about the existence of variables in different sets

Is it true that $$\mathbb A\subseteq \mathbb R,\: \mathbb A\neq \varnothing,\:\mathbb A\neq \mathbb R\implies \exists x,y\in\mathbb R : \begin{cases}x\not\in\mathbb A\\y\in\mathbb ...
0
votes
1answer
26 views

Closure Question in Enderton's 'Elements of Set Theory'

I am currently working on a follow-up question to the one I did here: Closure question from Enderton's 'Elements of Set-theory' I am unsure though whether I am on the right track with the ...
0
votes
1answer
26 views

Closure question from Enderton's 'Elements of Set-theory'

I am working on the following question but am unsure how to prove that $C^{*} \subseteq C_{*}$. Any help and comments would be appreciated. For those with Enderton's text on-hand, the page number is ...
0
votes
2answers
56 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
0
votes
1answer
90 views

Question about proof on basis

I found this proof online, but I have a bit of trouble understanding it. Question: Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an ...
3
votes
1answer
176 views

Can I pull out an arbitrary element from a set?

Suppose there is an onto function $f: \Bbb R \to \{0, 1\}$. I want to show that there is a function $g: \{0, 1\} \to \Bbb R$, such that $f(g(b)) = b$. I know that there are two element in the domain ...
2
votes
1answer
23 views

Union closure of a set of five finite sets

I've been playing with some basic set theory while looking at the Union-closed sets conjecture. Pretty basic question, but given four finite sets $A,B,C,D, \mathcal{Y}$ where $$A \cup B = C \cup D = ...
2
votes
1answer
58 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
1
vote
1answer
74 views

Question about proving that a finite intersection of big unions is a big union of finite intersections

Let $I_{1}$,...$I_{k}$ be index sets and for each $1 \leq m \leq k$ and each $j \in I_{m}$, let $U_{j}$ be a set. Prove that: $$(\bigcup\limits_{j_{1}\in I_{1}}U_{j_{1}}) \cap ... ...
0
votes
3answers
53 views

the use of Cartesian product

I am currently studying in secondery level. I read cartesian product the other day and I found it absolutely bizarre. In my text book, there is this "order pair" which I understood fairly well and ...
0
votes
2answers
20 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
0
votes
2answers
44 views

Under these assumptions, how many men play volleyball? [on hold]

There are 70 men in a club. Each plays at least one of the following games: volleyball, basketball and table tennis. 20 play volleyball only, 10 play basketball only and 6 play table tennis only, 4 ...
3
votes
4answers
76 views

Book/Article recommendation

I am a first year Math major in the university, this summer I want to self study and go over some specific subjects. Firstly, can someone can give a suggestion for a detailed book/article about the ...
0
votes
1answer
12 views

Simple question about indexing edges of an undirected graph.

As far as I understand, for an undirected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$, the set of edges is defined as unordered 2-element subsets of $\mathcal{N}$. So, for example, $\mathcal{E} = ...
0
votes
4answers
56 views

Can we write programs for all functions if we have an infinite alphabet?

If we have a finite alphabet, then the set of programs we can write is countably infinite (aleph naught). The set of all functions is uncountably infinite (cardinality of real numbers). If we have ...
1
vote
3answers
62 views

Cantor diagonalization and fundamental theorem

Can the Cantor diagonal argument be use to check countability of natural numbers? I know how it sounds, but anyway. According to the fundamental theorem of arithmetic, any natural number can be ...
1
vote
1answer
77 views

Proving the properties of big union of unions for indexed sets

Let $I$ be an index set, and for each $i \in I$, let $J_{i}$, be another index set. For each $i \in I$ and $j \in J_{i}$, let $U_{j}$ be a set. Set X = $\bigcup\limits_{i\in I}J_{i}$. Prove that: ...
2
votes
2answers
25 views

Proof of: $X$ is finite $\iff X$ is Tarski-finite

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876). In the fourth chapter, he deals with different definitions of finite set. Here is the classical one: ...
0
votes
1answer
79 views

question about Herbert B. Enderton's book : A mathematical introduction to logic

I hope someone can help me. My question arises on page 114 of the second edition of the book. Here the notion of 'prime formula' is introduced to enable one to view a formula as a formula of ...
2
votes
1answer
39 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
2
votes
1answer
50 views

Proof that there is a bijection, if there are injective maps in both directions

Let $A$ and $B$ be two sets. Let $f:A\to B$ be injective such that $Im(f) \subsetneq B$. Let $g:B\to A$ be injective such that $Im(g) \subsetneq A$. Obviously $A$ and $B$ are not finite sets. Can ...
8
votes
3answers
740 views

In Cantor's Diagonalization Argument, why are you allowed to assume you have a bijection from naturals to rationals but not from naturals to reals?

Firstly I'm not saying that I don't believe in Cantor's diagonalization arguments, I know that there is a deficiency in my knowledge so I'm asking this question to patch those gaps in my ...
2
votes
1answer
57 views

Prove this result about construction of sets

In Enderton's book on Set Theory, the following problem is given after introducing the notion of sets as an infinite hierarchy (I hope this much explanation is sufficient; if not, please mention and ...
3
votes
2answers
41 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
2
votes
5answers
52 views

How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable

My thoughts: If $A$ is denumerable then it has a bijection with $\mathbb{N}$ So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite. So if a single object is ...
1
vote
1answer
30 views

Divide Notation for Sets?

In the book "Abstract Alegbra" by Dummit and Foote, on page 260, problem 41c states: ...
1
vote
1answer
83 views

Alternative set therories?

Is there a version of set theory that allows the existence of a set that does not admit the empty set as a member? I.e., reject the axiom $A\cup \emptyset = A$
3
votes
3answers
471 views

Does there exist two distinct set which are not an element of a given infinite set?

Let $X$ be an infinite set. Under ZFC, it is true that $X\notin X$. So there is at least one set which is not an element of $X$. Is there another element distinct from $X$ which is not a member of ...
1
vote
4answers
59 views

Bijection between $\mathbb N^+ \times \mathbb R^+$ and $\mathbb R^+$

Let $\mathbb N^+$ denote the set of natural numbers bigger than $0$ and let $\mathbb R^+$ denote the set of real numbers bigger than $0$. Is there a way to write down an explicit bijection between ...
1
vote
0answers
42 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
0
votes
0answers
31 views

Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...