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, ...

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1answer
31 views

Calculate the cardinal of $\{f:\Bbb{N}\to \mathcal{P}(\Bbb{N}):n \notin f(n)\}$

Calculate the cardinal of $A=\{f:\Bbb{N}\to \mathcal{P}(\Bbb{N}):n \notin f(n)\}$. Ok, so I'm having some trouble with this problem. I know that since $A\subset \{f:\Bbb{N}\to \mathcal{P}(\Bbb{N})\}$ ...
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0answers
38 views

How to compute and compute the follow questioners about () and []

Compute the intersection of all sets of the form $(0, b)$, for $b$ a positive real number Compute the union of all sets of the form $[a, 1]$, for $a$ a positive real number Really don't know how to ...
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1answer
35 views

A bijection $f:\mathcal{A} \to \mathcal{B}$, where $\mathcal{A} \subset \mathbb{R}^n, \mathcal{B} \subset \mathbb{R}^k$, and $k <n$?

The motivation for this post is the question of whether or not there exists a class of functions which are bijections between vector spaces of differing dimensions. Although it is simple enough to ...
1
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1answer
22 views

Simple Set Operation with Random Variable

Consider $X(\omega) \ge -1$ be a discrete random variable and define an event $$ \{\omega: 1+a X(\omega) \le \varepsilon\} $$ where $a \in [0,1]$ and $\varepsilon \in [0,1]$. I was wondering ...
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2answers
35 views

Proof that for subsets $A$ and $B$ of a metric space $(X,d)$ that $(A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}$

Where the circle is the interior of the set. It seems quite trivial but I cannot find a direct proof for this: $(A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}$. Would this also imply that $A^{\circ} ...
2
votes
4answers
116 views

$|\mathbb N^{\mathbb N}| = |2^\mathbb N|$

I was trying to find a direct proof that $|\mathbb N^{\mathbb N}| = |2^\mathbb N|$, by finding a bijection between the two sets. The idea that came to mind was to start with the sequence of natural ...
2
votes
3answers
67 views

How to prove $A=(A\setminus B)\cup (A\cap B)$ [duplicate]

How to prove $A=(A\setminus B)\cup (A\cap B)$. I have seen this problem and the solution is clear to me. Initially I was satisfied by my prove but now I think it is wrong. How I have proved ...
4
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3answers
733 views

Does same cardinality imply a bijection?

This came up today when people showed that there is no linear transformation $\mathbb{R}^4\to \mathbb{R}^3$. However, we know that these sets have the same cardinality. I was under the impression ...
2
votes
1answer
71 views

order of infinite countable ordinal numbers

I'm trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the ...
0
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1answer
11 views

Question on M-generic filter

Let B a complete boolean algebra and $b, c\in B$ and M a model of ZFC. Why do we have that if $c\in G\,\, \forall G$ M-generic ultrafilter such that $b\in G$ then $b\le c$ ?
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1answer
65 views

What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms? [closed]

What expressive power of ZF theory, if any, would be lost if the Axiom of Infinity was replaced by some formal variant of Peano's Axioms to define the set of natural numbers?
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1answer
14 views

Simple question on predense set in a boolean algebra

Let B a complete boolean algebra and D a subsets of B. Then D is predense below $ b\in B $, i.e. the downward closure of D is dense below b, iff $b\le \bigvee D$.Proving this equivalence seemed like ...
1
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1answer
75 views

The value of limsup and liminf of a sequence of a sets obtained by combining three sequences

What is the limit superior of the following sequence of sets? $\{X_n\}=\{\{1/2\},\{1/3\},\{1/4\},\{2/3\},\{1/3\},\{1/5\},\{3/4\},\{1/3\},\{1/6\}......\}(n\to∞)$ I.e., $X_1=\{1/2\}, X_2=\{1/3\}, ...
1
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1answer
80 views

Let U be a universe. Use an element argument to prove the following statement. For all sets A and B in P(U), (A ∩ B)⊆(A U B).

I am currently stuck attempting this question. The only way I know how to solving this question is simply via (A ∩ B)⊆A and A⊆(A U B). Therefore (A ∩ B)⊆(A U B). However, as mentioned in the question ...
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2answers
30 views

When is the inverse of a reciprocated function equal to the function?

If $f(x)=f(-\frac{1}{x})$, are there finite or infinite solutions for this? Can we tell? Thank you very much.
2
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1answer
57 views

On the equality of two sets (a doubt from Probability with Martingales).

Let $(S, \Sigma, \mu) $ be $([0,1], \mathcal{B}[0,1], Leb)$. Let $\epsilon(k)$ be a sequence of strictly positive numbers s.t. $\epsilon(k) \downarrow 0$. Let $V = Q \cap [0,1],$ the set of rationals ...
3
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0answers
47 views

How are functions written as ordered triples?

In a function, I know that domain $A$ and codomain $B$ are not restricted to sets. They can be proper classes. In that case, how can we write functions as ordered triple? I used to think that in order ...
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3answers
72 views

Show that there exists a bijection from a sets that is countable and infinite into natural numbers.

Show that there exists a bijection from a sets that is countable and infinite into natural numbers. I know that this question is a bit dumb but I can't prove it explicitly. I mean I can't ...
2
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1answer
22 views

Question about boundary of set defined by a polynomial

Suppose $p(z):\mathbb{C}\rightarrow\mathbb{C}$ is is a polynomial of degree $n$. Define $M=\{z\, |\, \text{Re}\,p(z)<0\}$. Why is $\partial M=\{z\, |\, \text{Re}\,p(z)=0\}$? I have only read some ...
1
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1answer
23 views

negation of a null sequence

I have that a sequence $\{a_n\}$ is null $\Leftrightarrow \forall \epsilon >0, \exists X$ such that $$|a_n| < \epsilon \ \forall n > X.$$ I want to give a definition when a sequence is not ...
2
votes
1answer
22 views

The Union of $n$ Independent Events Equals the Complement of the Complement of Their Product

The Statement of the Problem: If the events $A_1,...,A_n$ are independent, show that $$ P\left(\bigcup_{i=1}^n A_i \right) = 1-\prod_{i=1}^n P(A_i^c) .$$ Where I Am: So, I've seen this equality ...
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3answers
87 views

Prove that rational numbers (not just positive) are countable without using axiom of choice.

Prove that rational numbers (not just positive) are countable without using axiom of choice(since it is controversial). I have seen proofs that use the fact that union of countable sets is countable, ...
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4answers
58 views

Proving that $A\setminus B\subseteq C$ implies $A\setminus C\subseteq B$ .

Theorem. Suppose $A, B$, and $C$ are sets, and $A\setminus B\subseteq C$. Then $A\setminus C\subseteq B$ . What I tried Proof: My try: From the statement $A\setminus C\subseteq B$ We ...
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2answers
22 views

Proving a set releation

Theorem:Suppose $A$, $B$, and $C$ are sets, $A\setminus B ⊆C$, and $x $ is anything at all. If $x ∈ A\setminus C$ then $x ∈ B$. Proof: Suppose $x ∈ A\setminus C$.This means that $x ∈ A$ and $x\notin ...
1
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1answer
67 views

constructions of terms using variable

It is usually said given set of variables, terms of language are defined recursively. But for recursive definition on a set, we need a function p which assigns to each function from a section of ...
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2answers
342 views

Impossible numbers drawn from tricky function

The function is this: $f(\frac{a}{b},\frac{c}{d})=\frac{a+c}{b+d}$ where $0\lt \frac{a}{b} \lt 1$ $0\lt \frac{c}{d}\lt 1$ $a,b,c,d$ are all integers $a/b$ and $c/d$ are in lowest terms Are there ...
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3answers
82 views

Proof of recursion theorem

I was going through a real analysis textbook The Real Numbers and Real Analysis this morning, and I encountered a theorem stating that: Let $H$ be a set, let $e\in H$ and $k:H\rightarrow H$ be a ...
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2answers
117 views

substituting a variable in a formula (in logic)

What kind of mathematical object is this substitution(is it a function or what). We assuming set of variables exist.
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2answers
21 views

Partition of at most countable set is at most countable set

Let $X$ be any at most countable set. I have to prove that partition of at most countable set is at most countable set. It seems bvious, but I need formal proof. Let $\{S_i\}_{i\in I}$ be partition ...
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2answers
52 views

Set Theory (Example of Set)

A set is defined as the collection of well-defined and distinct objects. Now if we consider the collection of identical glasses, can we call that collection as a set? But I am confused that since all ...
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3answers
58 views

Exercise 1 on page 10 in Naive Set Theory, following Axiom of Pairing

In Section 3 "Axiom of Pairing" in Naive Set Theory on page 10 Halmos proposes the following argument. "Consider the sets $ \emptyset , \lbrace \emptyset \rbrace , \lbrace \lbrace \emptyset \rbrace ...
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2answers
84 views

The range of the function $f(x,y)=(x+y,xy)$

I have the following homework question: $$\begin{split} f: \mathbb I \times \mathbb I &\to \mathbb R\times \mathbb R\\ f(x, y) &=(x+y, xy)\end{split}$$ Does there exist $(x, y) \in \mathbb ...
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2answers
39 views

Set Theory Proof $A=B$ [closed]

Let $A$ be the set of all integers $x$ such that $x = 2k$ for some integer $k$ Let $B$ be the set of all integers $x$ such that $x = 2k+2$ for some integer $k$ Give a formal proof that $A = B$.
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1answer
35 views

Size of Totally Ordered Set with Countable Predecessors

Assume Choice. Let $S$ be a set, and $\trianglelefteq$ be a total order on $S$. If for all $s \in S$, the set $\{t:t\trianglelefteq s\}$ is countable, what are the possible cardinalities of $S$? ...
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2answers
55 views

How do I write this set notation correctly?

I have a 2D matrix $f(m,n)$, where $1<m<M$ and $1<n<N$. The element at location $(m,n)$ is denoted as $f_{m,n}\in \{+1,-1\}$. I want to write something like set notation but I am not ...
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0answers
24 views

Prove/Disprove existence of a set [duplicate]

I would like to know if my arguments are correct. Prove/Disprove existence of a set X $\subset$ P($\mathbb{N}$) , $ |X|=\aleph$ and for every $ A,B \in X $ ,$ A \subset B $ or $ B \subset A$ I ...
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1answer
27 views

Let $R$ be an equivalence relation: How many elements are in $R$?

Let $R\subset X\times X, |X|=30$. Supposing there are only 3 distinct equivalence classes and all of these have the same amount of elements, find $|R|$. I didn't get very far on this, I thought that ...
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4answers
211 views

$\{∅\} ∈ \{∅\}$ is this right or wrong?

I am very confused about whether $\{∅\} ∈ \{∅\}$ or not. I thought it's right because in curly brackets the phy is also a member. Can anyone help me understand this?
2
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1answer
34 views

Manipulating sets ($+$, etc).

I was seeing a proof of the Open Mapping Theorem, in Kreyszig's book, and I have no problems with it. But there's a point in which he does something like: $$\begin{align}B_Y(0,r) \subset ...
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2answers
34 views

Comparison of two collections of 4-tuples using combinatorics - more complicated version

My problem is to show that 2 collections of unordered 4-tuples - $\mathbf{A}$ and $\mathbf{B}$ - are the same. I define a collection of objects as a set, in which multiple entries of the same object ...
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2answers
221 views

Showing a function is invertible

I came across this problem and not sure how to prove it. Show that if $ f\circ f \circ g\circ g \circ f\circ f $ is invertible then $ g $ is invertible. I'm not sure if it's correct to say that ...
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3answers
62 views

Why is the probability multiplied by $\binom{n}{k}$

A while ago I asked a question about probability here Why is binomial probability used here? I get that you can find how many ways of choosing the $6$ correct out of $10$ questions. But why do we ...
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2answers
41 views

Union of disjoint countable sets is countable [duplicate]

Suppose that $S_1$ , $S_2$ are disjoint countable sets of T .Then their union is countable ATTEMPT Let $S_1$ = ${x_1 ,x_2,...}$ $S_2$ = ${y_1,y_2,...}$ I am thinking of making pairs by doing ...
1
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1answer
23 views

Does all collection of sets have an index-set?

I have this form of the axiom of choice: Suppose that C is a collection of nonempty sets. Then thre exists a function $f:C\rightarrow \cup_{A\in C}A$ such that $f(A)\in A$ for each $A \in C$. ...
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5answers
137 views

Why can a closed, bounded interval be uncountable?

From what I have read, all finite sets are countable but not all countable sets are finite. As I understand it, Countably Finite --- a one to one map onto $\Bbb{N}$ with a limited number of members ...
2
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2answers
98 views

Can I soundly define a function which maps to itself?

A function can be defined by specifying a set of tuples. If I write the definition of a function $f = \lbrace(0, f) \rbrace$, is this function sound? Will this lead to a paradox? The domain of this ...
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1answer
38 views

Algebraic numbers as roots of polynomials of degree $n$ are countable.

For fixed $n \in \mathbb{N}$, let $A_n$ be he set of all algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$. Proof that $A_n$ is countable. (Hint: For ...
3
votes
1answer
47 views

Set Notation with exponent

I am looking at the function: $$f: \{5\}^2 \to \{5\}$$ it is certainly nothing too exceptional , but I find it difficult to understand what $\{5\}^2$ as a set notation and from then the whole ...
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4answers
1k views

Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
3
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1answer
52 views

Order for sets in the real line

Consider the sets $[0,1]$ and $[1,2]$. I want to say that $[1,2]$ is greater than $[0,1]$. Is there a set order such that $$A \geq B \quad \text{if} \quad \inf A \geq \sup B.$$ What is the name of ...