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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0
votes
1answer
17 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
votes
0answers
30 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" ...
2
votes
0answers
39 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 ...
3
votes
2answers
23 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 ...
5
votes
1answer
26 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 ...
0
votes
2answers
47 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
34 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
73 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
334 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
33 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
117 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
votes
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 $f \implies b=c$
3
votes
4answers
50 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
66 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
vote
1answer
43 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
vote
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
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} ...
3
votes
3answers
58 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 $|X| = 2^{2^{\aleph_0}}$.
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
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 ...
-10
votes
1answer
40 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
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 ...
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
45 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
50 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 $\mathcal{P}_1$ ...
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
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 ...
-2
votes
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 ...
-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
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
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 ...
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 ...
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
482 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 ...
1
vote
2answers
32 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 ...
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 $S=\{a,b\}$? [closed]

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
39 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
30 views

How do I prove this assertion? [duplicate]

Let $A$ be countable union of countably infinite sets. Then $A$ is also countable.
-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 ...
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
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 ...
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 ...
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 $(x,y) = (x',y') \iff x=x'\text{ and }y=y'$. ...