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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0answers
11 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 $A \geq B$ if $\inf A \geq sup B$ What is the name of such order?
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0answers
42 views

Explicit bijection $\Bbb R^{\Bbb R} \to P(\Bbb R) $.

Is there any simple way to construct a bijective function: $\Bbb R^{\Bbb R} \to P(\Bbb R) $ to see that $\Bbb R^{\Bbb R}$ is isomorphic to $P(\Bbb R)$?
1
vote
1answer
39 views

Set theory formula

I picked up a copy of Jech's Set Theory at my school library and I'm reading through it and taking notes. Right at the beginning, though, he mentions something called a 'formula'. Here's the quote: ...
0
votes
0answers
53 views

Set Theory (Real Numbers)

I have seen in a book that a number whose square is nonnegative is called real number. How can we explain what a real number is?
3
votes
5answers
70 views

What is a set of bijections?

I am taking a course on abstract algebra, and the lector defined $T$ to be a set, and defined $G$ to be the set of all bijections from $T$ to itself: $$ G=\{\text{all bijections }g\colon T\rightarrow ...
1
vote
3answers
38 views

Why is Binomial Probability used here?

A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. What is the ...
0
votes
3answers
30 views

Given two specific sets, show that one is a subset of another

Given $$X = \{x : x = 4^n-3n-1 ; n\in\mathbb{N}\}$$ and $$Y = \{y : y = 9(n-1); n\in\mathbb{N}\}$$ Prove that $X \subset Y$. I've been struggling with this problem for hours but I couldn't find a ...
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votes
1answer
42 views

Proving that $\mathbb{N}$ (or any countable set) is infinite

I was watching real analysis lectures by Francis Su. In the lecture on countable and uncountable sets, he writes a theorem which is my question: Thm. $\mathbb{N}$ (or any countable set) is ...
0
votes
2answers
36 views

Comparison of two sets of 4-tuples using combinatorics

My problem is to show that $\mathbf{A} = \mathbf{B}$. Specifically that $\forall a \in \mathbf{A} \implies a \in \mathbf{B}$ and $\forall b \in \mathbf{B} \implies b \in \mathbf{A}$, to be precise. ...
0
votes
2answers
78 views

Set Theory (Definition of a set)

A set is defined as the collection of well-defined and distinct objects. This implies that a member of a set can't be repetitive in the set. Now when we discuss groups in Group Theory, if we check the ...
0
votes
1answer
18 views

$R$ well-founded relation and $\forall y$, $\{x:xRy\}$ is finite implies $\forall y$, $\{x:xR^t y\}$ is finite (where $R^t$ is the transitive closure)

I am interested in proving the titular claim: $R$ well-founded relation and $\forall y$, $\{x:xRy\}$ is finite implies $\forall y$, $\{x:xR^t y\}$ is finite (where $R^t$ is the transitive closure) ...
2
votes
1answer
51 views

Does the Russell Set exist?

I am currently reading "Naive set Theory" by Paul Halmos. In the second chapter, on the axiom of specification we show that the Universal Set does not exist. The proof is the following: Lets ...
2
votes
2answers
43 views

What's the meaning of an element that belongs to the same element?

In classical set theory, if I consider that $x$ is an element, which means it is not a set, can I write $x \in x$ ? If yes, what this would mean? Correct me if I am wrong, but I don't need to have ...
1
vote
1answer
16 views

Proving statement about power sets using fact that Z = X ∩ Y

The statement I'm trying to prove is If Z = X ∩ Y , then P(Z) = P(X) ∩ P(Y). My proof is as follows: Let $U\in P(Z)$ so $U\subset Z$ and since Z = X ∩ Y, then $U\subset (X\cap Y)$. Hence ...
1
vote
2answers
35 views

If $A$ is a set and $\mathcal B$ is a set of sets, is there some shorthand for $\left\{A\times B:B\in\mathcal B\right\}$?

Let $A$ be a set and $\mathcal B$ be a set of sets. Suppose we want to define $$M:=\left\{A\times B:B\in\mathcal B\right\}\;.$$ Is there some shorthand for $M$ as we've got for $$X\times ...
0
votes
1answer
32 views

Have I actually shown that $X\subset Y$?

So I'm trying to show that $$If\ X\cup Y = Y\ then\ X \subset Y$$ I've had a go at a proof but I'm not sure if it actually proves the above at all: $$Let\ x\in Y$$ $$x\in (X\cup Y)$$ $$x\in X\ or\ ...
0
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0answers
32 views

$ \# \mathbb{R}^2 \geq \# \mathbb{R}$? [duplicate]

Well, I'm undergraduate in Math and was thinking about the following question: the cardinality of $\mathbb{R}^2$ is greater or equal to the cardinality of $\mathbb{R}$ (I believe it is not "less then" ...
3
votes
0answers
49 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
4
votes
2answers
31 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
6
votes
1answer
43 views

If a set $S$ has a choice function, does $\bigcup S$ have one too?

I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have ...
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votes
2answers
62 views

Powerset with constraints

I have two sets $NUMBERS$ and $LETTERS$ with: $ NUMBERS = \{1, 2, 3, 4, 5\} \\ LETTERS = \{ A, B, C, D, E\}$ No I want the power-set of my sets, i.e. the set of subsets of elements from both ...
2
votes
2answers
41 views

If $n < \aleph^*(m)$, then $n < 2^m$.

Without $AC$ Let $\aleph^*(m)$ be the least aleph that $\not\leq^* m$. I need a help or hint that if $n < \aleph^*(m)$, then $n < 2^m$. $a \leq^* b$ means we can define a surjective map from ...
7
votes
4answers
81 views

Showing a function $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ is injective

Let $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ with $$ f(i,j) = \frac{(i+j-2)(i+j-1)}{2}+j. $$ I want to show $f$ is an injection. This is how I approached the problem: I tried to show ...
5
votes
7answers
345 views

Properties that are true for finite sets but are (non-trivially) false for infinite sets

The finite analogue of the axiom of choice is true, and it seems highly intuitive that it would be true for the infinite case. It is, however, undecidable. When explaining this to myself or to others, ...
1
vote
1answer
27 views

Hilbert's hotel prime powers method

To fit an infinite number of coaches each with an infinite number of passengers, we can assign the people in the hotel with the prime number 2, and coach $c$ is assigned with the $c$th odd prime ...
1
vote
0answers
34 views

Disjoint set sum problem.

Let us have a set, denoted by $T$, and assign each element a position starting from zero, for e.g. in the set $T=\{1,2,3,4\}$, the positions are $T[0]=1,T[1]=2,T[2]=3,T[3]=4$. Also let's denote total ...
5
votes
2answers
118 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
-2
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0answers
45 views

How to prove that composition of functions is a function [on hold]

Using the fact that a function is a relation, which is a subset of the product of $X$ and $Y$. $(a,b)$ belongs to $f$ and $(a,c)$ belongs to $f \implies b=c$
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4answers
51 views

What does $f^{-1}(B)= \{ x \in X \mid f(x) \in B\}$ mean?

I have encountered the expression $$f^{-1}(B) = \{ x \in X \mid f(x) \in B\}$$ My questions are: 1) What does the $-1$ exponent mean in this context? 2) Is it right to say "if the set $X$ ...
4
votes
2answers
32 views

Prove that $f(X\cap f^{-1}(Y))=f(X)\cap Y$

Let $\ f\colon A\to B$ and let $X\subset A$, $Y\subset B$, prove that $$f(X\cap f^{-1}(Y))=f(X)\cap Y$$ The "$\subset$"$-$inclusion is easy: if $y\in f(X\cap f^{-1}(Y))$, exists a $x\in X\cap ...
2
votes
2answers
68 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
1
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1answer
44 views

formal definition of ordinal addition by recursion

I'm reading Kunen's Set Theory, An Introduction to Independence Proofs (1980). On page 26 he explains how to introduce ordinal addition through recursion. For the sake of convenience i'll give the ...
1
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1answer
65 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 ...
4
votes
5answers
63 views

Explanation of $\overline{\lim} A_n$ and $\underline{\lim}A_n$

Let $(A_n)_n$ be a countable family of subsets of a set $X$. We define: $$\lim \inf A_n = \underline{\lim} A_n = \bigcup_{n \in \mathbb N} \bigcap_{k \ge n} A_k$$ $$\lim \sup A_n = \overline{\lim} ...
3
votes
3answers
60 views

Transitive Closure of a Well-Founded Relation is Well-Founded (without Axiom of Choice)

I am interested in proving the titular claim: Transitive Closure of a Well-Founded Relation is Well-Founded (without Axiom of Choice) My approach: Let $R$ be a well-founded relation. We ...
4
votes
4answers
85 views

Is it possible to assign probability to a set $X$ with $|X|>2^{\aleph_0}$?

Is it possible to assign probability to a set $X$ with cardinality $|X| > 2^{\aleph_0}$? Example would be a set $|X| = 2^{2^{\aleph_0}}$.
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2answers
44 views

What does $A^{B}$ mean? [duplicate]

Assume, that A and B are finite sets. What notion $$A^{B}$$ does mean? Have been looking for awhile now.
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votes
3answers
24 views

If A and B are disjoint and B and C are disjoint so $A\cup C$ and B are disjoint

Prove: If A and B are disjoint and B and C are disjoint so $A\cup C$ and B are disjoint We know that $A\cap B=\emptyset \wedge B\cap C=\emptyset \rightarrow (A\cap B)\cap (B\cap C)= \emptyset ...
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votes
1answer
41 views

SET; RELATIONS; FUNCTIONS [closed]

For set of Real no. R which statement is false. (A). N is subset of R (B). $(a,b)$ is subset of R. $a<b$ (C). $\pi$ (pi) does not belongs to R (D). $\Phi$ (phi) is subset of R
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2answers
45 views

Question regarding proof that $V = \{ f : \Bbb N \to \Bbb N \mid f(n)\text{ is a prime for all }n \in N\}$ is uncountable

I'm studying for an exam for tomorrow and one of the old exams has this problem: Given the set $V = \{ f : \Bbb N \to \Bbb N \mid f(n)\text{ is a prime for all }n \in N \}$ Prove that this set is ...
1
vote
1answer
52 views

Power set of $\{\emptyset,\{\emptyset\}\}$ [duplicate]

For writing the power set of $\{\emptyset,\{\emptyset\}\}$, do I have to consider $\emptyset$ as null set or as a member of the given set? If I consider $\emptyset$ as a member, then the power set is ...
1
vote
1answer
46 views

Cartesian product with all elements

I have two sets A and B with $A = \{1,2,3\} \\ B = \{ A, B, C, D, E \}$ Now I need to get something similar to the Cartesian product. If my understanding is correct, the Cartesian product would ...
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1answer
26 views

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Answer: A ={$x|x^2+1=0$}.Explain like i'm five.

Write all elements of A.A = {$x|x^2<x<10$,x is a whole number}. Given Answer: A ={$x|x^2+1=0$}. Is this a typo?
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2answers
60 views
+100

Does meet of two partitions of a set always exist?

Let $\Omega$ be any set. Let $\mathcal{P_1}$ and $\mathcal{P}_2$ be partitions of $\Omega$. By $P_i(\omega)$ we denote cell of partition $i$ containing $\omega$. Meet of partitions $\mathcal{P}_1$ ...
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2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?
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2answers
47 views

An injection from R × {0, 1} to R [closed]

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
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1answer
42 views

Defining exponentiation on the integers

If one defines the integers as equivalence classes of pairs of natural numbers, there is a (canonical?) way to define addition and multiplication for the integers based on addition and multiplication ...
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0answers
34 views

“Elementary Set Theory - Leung, Chen” - Solution manual? [closed]

I'm trying to study some ST on me own :-) ! I have found a very nice book with lots of problems but without any solution to the problems. Do you guys know whether someone have made a solution manual ...
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2answers
42 views

Problem on elementary logic and set theory

Let A and B be sets with B is a subset of A. Prove that A \ (A\B)=B. I start by saying that suppose x is in A \ (A\B). By definition, x is in A and X is not in (A\B) . However, x is not in A\B ...
2
votes
2answers
42 views

Relationship between completeness and well ordering (meta).

Here is the definition for completeness of the reals (there are many equivalent formulations but I am interested in this one); Completeness: Every non-empty subset of the reals which is bounded above ...