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)...

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2
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4answers
35 views

Injective map from one set to other

In the theorem of Schroder-Bernstein, it is assumed that, given two sets $A$ and $B$, and there is an injective map from $A$ to $B$ and an injective map from $B$ to $A$. It then concludes that there ...
1
vote
1answer
28 views

entry vs. component

When speaking about tuples, I used to say and write "The number $a$ is the first component of the tuple $(a,b,c)$, and $c$ is its last component". This all went well until I had to speak about ...
2
votes
1answer
42 views

Is this an open or closed set?

$S=\{5+\frac{(-1)^n}{n} \; : \; n \in \mathbb N\}$ According to my calculations this set has a lower bound of $4$ and an upperbound of $5$; however, since $4$ is reachable by the set it is a minimum ...
1
vote
2answers
18 views

Counting subsets of different sizes of a set

Let $A$ be a non-empty set, and $\mathcal{P}^*(A)$ denote the power set of $A$ excluding empty set. There is a natural equivalence relation on $\mathcal{P}^*(A)$: for $S_1,S_2\in \mathcal{P}^*(A)$...
0
votes
1answer
23 views

Set notation for unordered cartesian product

In the question unordered cartesian product an shorthand notation for the unordered cartesian product was discussed but without any standard notation. So my question is what would be the explicit ...
1
vote
1answer
22 views

$R$ be infinite comuttaive ring with unity , $M,N$ be modules over $R$ , $f:M \to N$ be surjective module homomorphism ; then $|M|=|N ||ker f|$?

Let $R$ be an infinite commutative ring with unity , $M,N$ be modules over $R$ , let $f:M \to N$ be a surjective module homomorphism ; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
3
votes
2answers
70 views

Why do we need axiom of choice for this?

The answerers on this question say that we need AoC (or some variant thereof) to prove that every infinite set has a countably infinite subset. In my view, choice is not needed but the answerers are ...
9
votes
5answers
231 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 ...
0
votes
1answer
26 views

Ordinal Numbers: is it possible to have an isomorphic copy of an ordinal $\alpha$ inside an ordinal $\beta < \alpha$?

Could there be a function $f:\alpha \to \beta$ order-preserving and injective such that $\alpha$ and $\beta$ are ordinal numbers with $\alpha > \beta$?
1
vote
3answers
39 views

How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way. What is a popular way? In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?
-2
votes
2answers
33 views

Simplify the following question [on hold]

Simplify the following expression: $$(A\cup B)\cap (A\cup C)$$
0
votes
2answers
24 views

If $f: A \rightarrow B$ is surjective, and $A, B$ are nonempty sets, and $X \subseteq A$, does $f(A) - f(X) = f(A - X)$?

I'm working on a proof, and the proof will be complete if this is true... but I can't find a theorem in my book that explains whether or not this is true.
0
votes
2answers
23 views

Question about dictionary orders for $(\mathbb{Z}_+^{\omega})$

I just want to make sure I understand the explanation stated below. According to the order relation stated below we have $(a_0,a_1,...) < (b_0,b_1,..)$ if $a_i = b_i$ for finitely many values ...
0
votes
2answers
42 views

Subtle difference between statemetns invloving negation in set theory.

What is the difference between the statements $x$ is not in an infinite number of sets $E_n$ and ...
3
votes
4answers
70 views

Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$ For ...
1
vote
3answers
98 views

Sum of finite ordinals: $\lambda_1+\lambda_2+\dots+\lambda_n+\dots=\omega$

Prove: $\lambda_1+\lambda_2+\dots+\lambda_n+\dots=\omega$, where $\lambda_i$ are finite ordinal nonzero numbers. I tried like this. $A_i$ set and ord($A_i$)$=\lambda_i, i=\{1,2,...\}$ and ord($\...
-4
votes
1answer
62 views

Is ω really the first ordinal transfinity? [on hold]

The Internet claims that ω is the first ordinal transfinity. But what about ω-1? Isn't that a ordinal transfinity, and isn't it before ω? Kind of like ω/2? I guess I lack an understanding of what ω ...
0
votes
0answers
42 views

Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"
2
votes
2answers
25 views

which set is including $k$

$A=\{x^2+k \mid x \in \mathbb Z,-3 \leq x<k\}$, where $k$ is a constant. If $\{6,9\}\subseteq A$, then which set below includes $k$? $\{5x+1\mid x \in \mathbb Z\}$ $\{4x+3\...
0
votes
1answer
82 views

Can $\bigcap_{B\in A}B=\emptyset$ Given that all elements of $A$ are inductive sets?

I am reading a course in mathematical analysis vol 1 by J.H. Garling. He defines a successor set as one that (1) contains $\emptyset$, and (2) contains $a^+$ whenever it contains $a$ (where $a^+$ is ...
1
vote
1answer
27 views

Proof-verification: $A\times B \subset C \times D \Rightarrow A\subset C$ and$B \subset D$.

I think I have a proof but Munkres' statement "assuming $A$ and $B$ are nonempty" is making me unsure. [EDIT to clarify: This statement is given twice, once without the "assuming nonempty" and once ...
0
votes
1answer
17 views

Finding equivalent statements with quantifiers

Find equivalent pairs: a. $\forall x(P(x)\land Q(x))$ b. $(\forall x(P(x))\land (\forall xQ(x))$ c. $\exists x(P(x)\land Q(x))$ d. $(\exists x(P(x))\land (\exists x Q(x))$ Are ...
1
vote
1answer
28 views

How do I formalize the topology generated by a subbasis?

The topology generated by a subbasis $\mathcal{S}$ is defined as the colection $\tau$ of all unions of finite intersections of elements of $\mathcal{S}$. I want to formalize $\tau$ as something ...
3
votes
1answer
44 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
2
votes
4answers
66 views

How do you prove that $p → q$ is equivalent to $p \lor q ↔ q$?

I gotta draw $p \lor q ↔ q$ from $p → q$, logically. not by a truth table. While it seems obvious, I cannot find a formal proof. This is how far I came up to: $\quad p \lor q$ $\equiv (p \land T) \...
2
votes
1answer
21 views

Prove that if $R$ is a symmetric, transitive relation on $A$ and the domain of $R$ is $A$, then $R$ is reflexive on $A$.

Assume, $R$ is a symmetric, transitive relation on $A$ and the domain of $R$ is $A$. $Dom(R)=A$ implies $(\forall x \in A)(\exists y \in A)[xRy]$. Since, $xRy$ is true it follows that $yRx$ is ...
3
votes
5answers
81 views

How are sets “detached” from their structure?

This question is best asked with an example. Consider the real numbers. However we construct the real numbers, the "final product" so to speak, is not just a set, but it is a complete ordered field. ...
2
votes
1answer
38 views

Describe the equivalence classes generated by T

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$. Question Describe the equivalence relation T on the real line that is the intersection of all equivalence ...
1
vote
4answers
231 views

Pronuntiation of the symbol $\varnothing$ of the empty set

The symbol $\varnothing$ for the empty set was introduced by Bourbaki, inspired by the Norwegian alphabet $\varnothing.$ It has no relation with the Greek letter $\phi.$ From my schooldays, when the ...
6
votes
2answers
201 views

Cardinality of the set of all infinite monotonically decreasing sequences of naturals

Find the cardinality of the set of all infinite monotonically decreasing sequences of naturals. I think it's $\aleph_0$. I marked this set in $A$, and said that $\forall n\in\Bbb N \ (n,n,n,...)\in ...
1
vote
2answers
22 views

Is there a faster way to determine partial orderings of basic finite sets?

For example, consider the set $S = \{ 0, 1, 2, 3 \}$, and the following relation on $S$: $$ R = \{(0,0), (1,1), (1,2), (1,3), (2,0), (2,2), (2,3), (3,0), (3,3) \}. $$ Obviously, I can go through ...
1
vote
2answers
33 views

Find an example such that $X$ with the lexicographic order is not well-ordered.

Let $\{A_n\}_{n\in\Bbb N}$ be a collection of well-ordered sets. $X$ is defined by $X=\prod_{n\in\Bbb N}A_n$. Find an example such that $X$ with the lexicographic order is not well-ordered. I know ...
4
votes
1answer
343 views

Epsilon numbers

Let $\alpha$ be an ordinal number and define $f_\alpha$ as: $f_\alpha(0) = \alpha + 1$ $f_\alpha(n+1) = \omega^{f_a(n)}$ Let $S(\alpha) = \sup\{f_a(n)\ |\ n \in \omega\}$ Then $S(\alpha)$ is an ...
2
votes
1answer
24 views

Intersection of a nested interval of $A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$

$A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$ What is $\bigcap_{n=1}^{\infty}A_n$ Since every set becomes a subset of the next set, is it correct to say that the intersection of all ...
0
votes
4answers
44 views

Find a counter example

The interior of the union is the union of the interiors. $\text{int}\left(A\cup B\right) = \text{int}(A) \cup \text{int}(B)$ I'm not too sure about to get started with this one. Any hints so as to ...
0
votes
1answer
25 views

Is $R$ an equivalence relation?

Let $X,Y$ be infinite sets. Define $F$ as $F=\{f:X\rightarrow Y\}$ . We define a binary relation $R$ on $F$: $fRg$ if there is no countable $S\subseteq X$ such that $\forall x\in S \ f(x)\neq g(x)$. ...
-2
votes
1answer
39 views

Are $(\mathbb R\times \mathbb Q, \le_h)$ and $(\mathbb Q\times \mathbb R, \le_h)$ isomorphic? [on hold]

Are $(\mathbb R\times \mathbb Q,\leq_h)$ and $(\mathbb Q\times \mathbb R, \leq_h)$ isomorphic? when "$\leq_h$" is the right lexicographic order? Thanks a lot!
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: $X_{1}=\emptyset$ ...
4
votes
1answer
48 views

Why is the axiom of choice controversial? [duplicate]

In other words, what are the arguments for ZF over ZFC, and what philosophical issues have people raised against including it as a standard axiom of set theory?
4
votes
1answer
4k views

If the complement of the universal set is the null set, then the null set is not in the universal set?

Let $\Omega$ be the universal set (which contains all objects of interest) and let its complement be denoted by $\Omega^c$ . The book I'm reading states that $\Omega^c = \phi$ ; does that mean $\phi \...
1
vote
0answers
24 views

Formalization of an intuitive idea to construct a surjection

Let $A$ be an arbitrary set and $B$ be any non-empty set. Furthermore, suppose that there is no injection from $A$ to $B$. I want to prove that it follows that there is a surjection from $A$ to $B$. ...
0
votes
1answer
31 views

Set Theory Introduction [on hold]

what does it mean that a set is element of itself if a set is element of itself I must add it one element and then that set is not the old one so contradiction
1
vote
1answer
32 views

Cantor's diagonal argument: Prove that $|A|<|A^{\Bbb N}|$

Let $A_1\subseteq A_2\subseteq A_3\subseteq...$ be a raising series of sets such that $\forall n\in \Bbb N \ |A_n|\lt |A_{n+1}|$. We mark $A$ as $A=\bigcup_{n\in\Bbb N}A_n$. Prove that $|A|<|A^{\...
0
votes
2answers
174 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
2
votes
1answer
59 views

Olympic Problem about Theory of numbers.

Let $Y=\{1,2,\ldots, 2014\} \subset \mathbb{N}$. Find the maximal subset $A\subset Y$ such that, $$\forall x\in A,\quad x\not\mid\sum_{y\in A\setminus\{x\}} y.$$ Example, $A'=\{2,4,6,\ldots,2014\}\...
0
votes
3answers
55 views

vacuous truth -> empty set is both included and not included in every set?

I understand the concept of vacuous truth and its use in showing that the empty set is a subset of every set. Based on my understanding of vacuous truth (for example https://en.wikipedia.org/wiki/...
1
vote
1answer
51 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 $100$...
3
votes
1answer
35 views

Boundary and Interior of set $\{-3,2,5\}$

I'm trying to see if I'm correctly understanding and applying the definition for interior and boundary points. Interior point: A point x in R is an interior point of S if there exists a ...
0
votes
0answers
20 views

How am I to understand this notation with regards to bdS and the int S? $S=\bigcap_{n=1}^{\infty}\left(-\infty,7+\frac{1}{n}\right]$

I'm trying to find the largest $\epsilon$ such that the neighborhood centered at $x$ of radius $\epsilon$ is contained in $S$. That part I think I can do, but I just don't know understand the below ...