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, ...

learn more… | top users | synonyms

1
vote
2answers
51 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 ...
0
votes
1answer
30 views

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$.

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$. I know this is a true statement. I am trying to prove if because I would also ...
0
votes
1answer
64 views

Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$

I want to prove this: $$(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$$ by every element on LHS(left hand side) is an element of RHS and vice versa. Does a counter example exist?
1
vote
1answer
40 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 ...
3
votes
1answer
128 views

Planar kelvin problem

What is the minimal possible value of the maximal total side length shared by any two tiles in a tiling of the plane if all tiles have the same area $A$? $\text{Total side length} = ...
3
votes
3answers
43 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
74 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 $|X| = 2^{2^{\aleph_0}}$.
2
votes
2answers
48 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 $\mathcal{P}_1$ ...
5
votes
3answers
716 views

Bijection between the reals and the set of permutations of the natural numbers?

In analysis today we talked about re-arrangements of sequences, and one student asked how many re-arrangements there are of a given sequence. We were able to very quickly create a one-to-one function ...
1
vote
1answer
59 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
61 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} ...
1
vote
1answer
86 views

Well ordering of type epsilon one

I have been very interested in the countable ordinals for awhile now, but one thing has eluded me despite my research into the subject. What is a well-ordering of the natural numbers corresponding to ...
9
votes
3answers
764 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
0
votes
2answers
43 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.
0
votes
3answers
23 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 ...
39
votes
9answers
3k views

Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use ...
-10
votes
1answer
39 views

SET; RELATIONS; FUNCTIONS [on hold]

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
11
votes
1answer
146 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
3
votes
2answers
44 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 ...
14
votes
3answers
8k views

Prove/Disprove that if two sets have the same power set then they are the same set

I am really sure that if two sets have the same power set, then they are the same set. I just am wondering how does one exactly go about proving/showing this? I'm usually wrong, so if anyone can show ...
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
40 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 ...
0
votes
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?
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 ...
0
votes
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 ?
-2
votes
2answers
47 views

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

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
33
votes
7answers
4k views

Why can't a set have two elements of the same value?

Suppose I have two sets, $A$ and $B$: $$A = \{1, 2, 3, 4, 5\} \\ B = \{1, 1, 2, 3, 4\}$$ Set $A$ is valid, but set $B$ isn't because not all of its elements are unique. My question is, why can't ...
0
votes
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 ...
3
votes
1answer
35 views

Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $ A $ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
-2
votes
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
0answers
33 views

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

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 ...
0
votes
1answer
19 views

Set intersection of finite,nested sets of real numbers [duplicate]

I'm currently trying to write up a solution to the following problem: If $ \displaystyle A_1 \supseteq A_2 \supseteq A_3 ... $ where each $ \displaystyle A_j $ is a non-empty, finite set of real ...
0
votes
1answer
19 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
5
votes
3answers
480 views

Finding a bijective function between an open disk and the open square

How can I find a bijective function between these two sets? $$\{(x,y)\in\mathbb{R}^2 \,|\, x^2+y^2<1\}, \quad (-1,1) \times (-1,1) .$$ I already thought of first writing between 2nd and set ...
2
votes
1answer
24 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
1
vote
2answers
31 views

How do I find the type of relation on an infinite set?

Imagine I'm given a set A = {∅, {∅}, {{∅}}, {{{∅}}},…} where ∅ is empty-set. Then I also have a relation on that set (actually on its power set) defined as: $R \subseteq \wp(A) \times \wp(A)$, where ...
0
votes
3answers
433 views

Family of union and intersection (set-theory)

If $S$ is the set of real numbers, and if $T$ is the set of rational numbers, let, for $\alpha \in T, \ A_{\alpha}= \{x\in S\ | \ x\ge \alpha\}$. Can anyone help explain why $\cup_{\alpha\in ...
1
vote
1answer
26 views

Find bijective correspondence between the sets

Find bijective correspondence between the set of all functions of $X$ in the set $\left\{ 0,1 \right\}$ and the power set of set $X$ and find $| 2 ^ X |$, if $| X | = n.$ My thoughts: ...
0
votes
1answer
39 views

How to show venn diagram?

How to show the following sets by Venn diagrams? Case 1: $$A=\{1,2,B\},B =\{3,4\}$$ Case 2: $$A=\{1,2,3,4\}, B=\{3,4\}$$
-2
votes
1answer
46 views

How many elements are in the set $S^S$, where $S=\{a,b\}$? [on hold]

If set $S =\{a,b\}$, then how many elements will be in set $S^S$? Here $S^S$ is {Set S is Exponent of S}. Do we need to do cross product like $S*S$ when it says $(S^S)$. Please advise.
0
votes
2answers
38 views

Proof in set theory

Let $A,B,C$ -- subsets in some fixed set. Prove that $A \cap B \subseteq C$ iff $A \subseteq \overline{B} \cup C$. Have no ideas how to prove this. On the language of definitions we have $$x ...
0
votes
0answers
29 views

How do I prove this assertion? [duplicate]

Let $A$ be countable union of countably infinite sets. Then $A$ is also countable.
1
vote
2answers
26 views

A simple way to know whether a well-ordered set has a subset of a certain type

Following my last question, Does $\Bbb R-\Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$, I would like to have better tools to look at a set and know what order types can it have. I ...
3
votes
1answer
55 views

Does $\Bbb R-\Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$

Does $\Bbb R- \Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$? I thought of taking the subset to be A={$n\cdot \sqrt{m}:n\in\Bbb N,m\in P$} where P is the set of all prime numbers, ...
-1
votes
2answers
33 views

Sets cardinality definition

I have a question about cardinality definition. How can we formally define cardinality for finite set using only maps from natural numbers to the set? UPD One says that the cardinality can be ...
3
votes
1answer
57 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
3
votes
3answers
115 views

(Is it a set?) Set of all months having more than 28 days.

Set is a well defined collection of distinct objects. Is the following is a set? Set of all months having more than 28 days. I'm confused here. Because on one hand I think that it is well ...
1
vote
1answer
56 views

Cantor-Bernstein Proof

Currently, I am studying Set Theory, and have come to the point of proving the Cantor-Bernstein Theorem (if $|A| \leq |B|$ and $|B| \leq |A|$, then $|A| = |B|$). Now, I am studying from Jech and ...
3
votes
1answer
35 views

Cardinality of all linear transformations from $\Bbb R^3$ to $\Bbb R^2$

I tried to calculate the cardinality of all linear transformations from $\Bbb R ^3$ to $\Bbb R^2$. This is my answer- I would like to know how to formalize it better. A transformation is defined in ...
0
votes
1answer
36 views

Problems with this Cartesian Product definition

Supposed I do not define ordered pair in the usual Kuratowski way $(x,y) = \{\{x\},\{x,y\}\}$. I left the ordered pair undefined but with the propriety $(x,y) = (x',y') \iff x=x'\text{ and }y=y'$. ...