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
0answers
24 views

Progression of Cardinality of powers of two

I am researching how to represent a progression of sets and their cardinalities. I can use some guidance on what I am looking for. There exists a progression of elements and sets such that the ...
0
votes
1answer
8 views

An indexed family of filters and their elements

Let $X$ is an indexed (by some set $n$) family of filters (on some poset $\mathfrak{A}$). Is there any standard notation/terminology for the set $\{ y\in \mathfrak{A}^n \,|\, \forall i\in n:y_i\in ...
1
vote
1answer
25 views

Does an order relation on a set induce an order relation on the power set?

Suppose $A$ is a simply ordered set. Is there a natural induced simple order on the power set $\mathcal{P}(A)$ ? If A happens to be well ordered the following seems to define a simple order on ...
0
votes
3answers
40 views

Proof that $X^C \cap Y^C= \;(X \cup Y)^c$

Proof that if $X \subset S,\; Y\subset S,\;$ then $\;X^C \cap Y^C= \;(X \cup Y)^c:$ It must be shown that the two sets have the same elements, that each element of the set on the left is an element ...
0
votes
2answers
46 views

Suppose $A, B$, and C are sets. Prove that $C\subset A\Delta B \Leftrightarrow C \subset A \cup B$ and $A \cap B \cap C = \emptyset $

The problem statement is in the title. I'm proving a problem in class and I need to show the above containment. I've drawn some Venn diagrams to make sure the containment makes sense, and it does to ...
0
votes
1answer
21 views

countable set that contains 1 and pi and has polynomial with coefficients in set s.t. all real roots are in set

Deduce that there is a countable set X that contains the real numbers 1 and pi and has the further property that if P is any non-zero polynomial with coefficients in X, then all real roots of P belong ...
1
vote
5answers
117 views

How can one find a set of given cardinality and disjoint from a given set?

In Algebra by Serge Lang, the author asserts, to prove the existence of a field extension where an irreducible polynom has a root, that if you take one set $A$ and a cardinal $\mathcal{C}$, that you ...
1
vote
1answer
43 views

$\cup_{i=1}^n [\frac 1n , 1] = ]0,1] $??

I was just confused, wheater $\cup_{i=1}^n [\frac 1n, 1] = ]0,1]$ is. I also thought that it might be [0,1] but I think that is not true. Cheers
1
vote
4answers
104 views

ZFC: Why is the set $\{ x \mid x = x\} $ not defined?

Why is the set $\{ x \mid x = x\} $ not defined? Since, $x=x$ is always true, the set is actually "the set of everything". But why is it illegal to be defined as a set?
0
votes
1answer
37 views

Bijection and image

Let f: A -> B be a bijection, so f^-1: B -> A is a function. Let X be a subset of A. How do I prove that Im(f)(X) = Preim(f^-1)(X)? Thank you.
2
votes
3answers
52 views

Determining injectivity and surjectivity

Are these functions injective or surjective? Also, how should I go about proving this? The function maps $ℕ×ℕ$ to $ℤ$. $f(a,b) = 4a+5b$ $f(m,n) = m^2-n$ $f(p,q) = 5^p·3^q$ Thanks!
2
votes
1answer
47 views

I currently know Calculus I — What steps would I take to understand Zermelo–Fraenkel set theory?

While this question can be discussed, it should have a clear answer by stating the following: How can one go from a high school / low-level college understanding of mathematics (completed Calculus ...
1
vote
1answer
16 views

Solving a poset for less than equal?

I don't completely understand posets yet, so I'm confused on how to do this particular problem. Here is the question: Let S be the set of all real numbers. Prove that the less than or equal to ...
1
vote
1answer
36 views

Equivalance relation [closed]

I wonder if there is something more general than the concept of the equivalance relation and in general the concept of relations .Any thoughts of how such a generalization is? Thank you!
0
votes
2answers
32 views

Does defining the closure of a set as the intersection of all closed set that contain it requires the axiom of choice?

Given a set $S$, the closure of $S$ is sometimes defined as the intersection of all the closed sets that contain it. This type of argument is pervasive in mathematics when one want to construct the ...
-1
votes
1answer
41 views

Prove $a^{\smile \smile} =a $ where $a$ is a set.

So far, I have this : (applying definition of $\smile$)$a^{\smile \smile} = \{(u,w) | (w,u) \in a\} ^{\smile}$ (applying definition of $\smile$) $a^{\smile \smile} = \{(i,j) | (j,i) \in \{(u,w) | ...
1
vote
2answers
30 views

Cartesian Product for sets

Recall that the Cartesian product $A\times A$ is defined as the set $\{(x,y) : x ∈ A \land y ∈ A\}$ . Thus if for example $A = \{1,2,3\}, A\times A = ...
0
votes
1answer
70 views

Surjectivity of Composite Functions

The question I'm asking might be rather simple, but I couldn't find relevant information (maybe it's too trivial?). Here's the question that baffled me. Let $f:X\rightarrow Y$ and $g:Y\rightarrow ...
0
votes
1answer
24 views

An issue with the usual proof of Cantor's Theorem

It seems to me that the standard proof of Cantor's Theorem also "proves" that $\left|\mathcal{P}(X)\right| < \left|\mathcal{P}(X)\right|$. [The following is adopted from Hrbacek & Jech.] ...
0
votes
1answer
31 views

Why does this collection is a set and what is this symbol?

My question is quite simple, I'm trying to understand why $Hom_{sets}(X,Y)$ is a set and what is this symbol $Y^X$? I'm sorry, I'm sure this should be a silly question, but I only know very basic ...
2
votes
3answers
88 views

What is the difference between $\Bbb{R^n}\times\Bbb{R^m}$ and $\Bbb{R^{m+n}}$?

My book specifies a function: $$\Bbb{R^n}\times\Bbb{R^m}\to\Bbb{R^m}$$ What is the difference between $\Bbb{R^n}\times\Bbb{R^m}$ and $\Bbb{R^{m+n}}$? And if there is none, what are the relative ...
0
votes
2answers
47 views

How does the empty set work? [duplicate]

I'm having trouble understanding what exactly an empty set is. Does ∅ mean "{}" ? and what is the difference between ∅ and {∅} ? If someone could shed some light on this, and provide a couple of more ...
1
vote
1answer
22 views

Is this set of functions countable?

I want to know if the set of functions $F=\{f:\mathbb{Z}\to\mathbb{Z}:f(n)\neq 0 \text{ for finitely many n}\}$. I haven't done a lot of progress really, but considering define the sets $A_n=\{f: ...
0
votes
6answers
55 views

Understanding the Empty set and set theory via proof [duplicate]

This question has already been asked. see above link.
-1
votes
2answers
30 views

Prove that there is bijection between sets

I need to prove that there is a bijection between these sets: $$A = [0, 1], B = (0, 1/2) ∪ (1/2, 1)$$ I tried to use Cantor–Bernstein–Schroeder theorem but I am lost. Can you help me?
0
votes
1answer
30 views

Prove that if there exists f: X to a finite set is bijective, then X is finite

I was given that X is finite if any function that maps X to X is surjective and injective. Also, the problem specifies the finite set n as a set with n elements. Now, I only know that there exists ...
1
vote
1answer
39 views

Is a non-empty set a partition of itself?

I have a homework assignment where, given some definitions, I need to prove that every set has a partition. There were some very elaborate ideas on how to prove this, but I realized that the ...
1
vote
1answer
68 views

Bigger infinity than real number infinity [duplicate]

Is there a bigger infinity than the infinity of cardinality of the real numbers $R$ ? i.e. is there a set to which real numbers can't be mapped one-one to ?
0
votes
3answers
52 views

Are the following sets countable?

I'm trying to determine if the following sets are countable: (a) $\mathbb{Z}^{[0,1]}, (b) [0,1]^{\mathbb{Z}}, (c) \mathbb{Z}^{\mathbb{Z}}$, (d) the set given by functions $f:\mathbb{Z}\to\mathbb{R}$ ...
0
votes
1answer
18 views

Bijection between sets with bounded difference.

Does there exist a bijection between the set of integers and the set of even integers if the absolute value of the difference between any two "paired" numbers cannot exceed one billion? I know that ...
2
votes
1answer
43 views

A couple questions about a countable set I encountered

Last week, we worked on a proof in class, but I'm having a hard time following several passages in the proof. ${}^\omega \omega$ denotes the set of functions from $\omega$ to $\omega$. At one point ...
3
votes
1answer
44 views

Why do some authors define the ordered pair as the set: $(a,b)=\{\{a\},\{a,b\}\}$? [duplicate]

I am using a textbook called foundations of mathematical analysis by johnsonbaugh and in it, he defines the ordered pair of elements $a$ and $b$, writen as $(a,b)$ as the set: ...
1
vote
2answers
47 views

Prove that $A=B$ considering: $(A \cap C = B \cap C) \land (A \cup C = B \cup C)$

So this is the thing I've got to prove: $ (A \cap C = B \cap C ) \land ( A \cup C = B \cup C ) \Leftrightarrow A = B $ I can understand this intuitively but the formal proof is taking me some time. ...
0
votes
2answers
40 views

Is every set with this property uncountable?

I have a non-empty set $\Gamma = \{(\rho, \sigma)| (\rho, \sigma) \in (0,1)^ 2\}$. Set $\Gamma$ has the following property: For any $(\rho, \sigma) \in \Gamma$ and any $(w,v) \in \mathbb{N}^2$, there ...
0
votes
1answer
23 views

Cardinal arithmetic basics

Let's say we have $\omega + \omega$. Since these sets are not disjoint we can replace them by disjoint sets of the same cardinality, namely $\omega \times \{0\}$ and $\omega \times \{1\}$. Then $\big ...
1
vote
2answers
35 views

Proof over subsets

So I'm currently taking a course for proofs, could you please check my work? Prove if $B \subseteq C$ then $A \cup C^c$ is a subset of $A \cup B^c$. For all $x \in B$, $x$ will be an element of $C$. ...
0
votes
2answers
27 views

How to solve this question subset

Answer true or false to each of the following questions. If a statement is true, prove it. If a statement is false, give a counterexample. For all sets $A$,$B$ and $C$: IF $A ⊆ B$ and $A ⊆ C$, Then ...
4
votes
3answers
393 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
3
votes
3answers
251 views

Is every set a pointed set?

My question is quite simple, It seems every non-empty set is a pointed set, only we have to do is choice some element to be the distinguished element, am I right? I'm looking for non-empty sets which ...
0
votes
2answers
23 views

Proving that if $E, F$ are equivalence relations on $A$ and $E \subseteq F$, then there is a surjection from $A\setminus E$ to $A\setminus F$

Proving that if $E, F$ are equivalence relations on $A$ and $E \subseteq F$, then there is a surjective function from $A\setminus E$ onto $A\setminus F$. What does $E \subseteq F$ even mean? Does it ...
-6
votes
1answer
79 views

Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
-2
votes
2answers
37 views

Set theory (A\C) \ (B\C) = (A\B)\C [duplicate]

Very difficult question. Can't make sides equal after multiple manipulations.This one is lost on me. Help a fellow human out. Thank you.
2
votes
1answer
32 views

What axiom makes it possible to take the union or intersection of an infinite number of sets $A_1, A_2, \ldots,$ and get a resulting set $B$.

What axiom makes it possible to take the union or intersection of an infinite number of sets $A_1, A_2, \ldots,$ and get a resulting set $B$. In probability I've calculated $P(\cup_{i=1}^{\infty} ...
1
vote
1answer
44 views

What's the negation of “E is uncountable” ??

I think "E is countable or finite." But when I asked my professor, he said "E is countable."
-1
votes
1answer
53 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
1
vote
3answers
37 views

Empty intersection of empty sets [duplicate]

In what sensible way can we define the union and intersection of an empty family of sets? How come empty intersection of empty sets become the whole space ?
1
vote
3answers
53 views

Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
0
votes
1answer
27 views

Cardinality of the union of all repeated Cartesian products of N with itself [duplicate]

Here is a silly question, but I am a silly person. Consider the: Natural Numbers. Natural Numbers X Natural Numbers. Natural Numbers X Natural Numbers X Natural Numbers ... Now take the union of all ...
2
votes
5answers
130 views

Help me understand 'equivalence classes' and relations

I'm reading up on binary relations and I understand them to be a mapping from one set into another. However I'm having problems understand 'equivalence classes'. My book only gives a pretty dry ...
7
votes
2answers
194 views

cartesian product $A^2 = A$, possible?

Do there exist non-empty sets $A$ such that $A\times A = A$? $A\times A = A$ looks a little strange to me, since $A\times A$ seems somehow more complicated than $A$, hence it is unlike that they are ...