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, (un)...

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2
votes
1answer
45 views

Range of $f\in C_c(X)$ is compact subset of complex plane

The collection of all continuous complex functions on $X$ whose support is compact is denoted by $C_c(X)$. Book's proof is quite not detailed and I will write a detailed proof. Proof: Let $f\in ...
2
votes
1answer
34 views

Uncountable subcollection of open sets with nonempty intersection

Given an uncountable collection of open intervals in $\mathbb{R}$, can we choose an uncountable subcollection which will have unempty intersection? I read that indeed, we could do that. But on the ...
1
vote
1answer
32 views

Any subset of elements in an ultrafilter is in the ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ an ultrafilter on $X$, suppose $ A \in \mathcal{F}$ and $B \subseteq A$, then either $B$ or $A\backslash B$ are in $\mathcal{F}$ ...
0
votes
0answers
96 views

An equivalence relation on Dedekind cuts

The definition of a cut $(A,B)$ is: $$ \mathbb{Q} = A \cup B \\ A \ne \emptyset ,B \ne \emptyset \\ \forall a \in A,\forall b \in B,a < b \\ $$ Define a relation on the set of all cuts of $...
1
vote
1answer
28 views

Given $\mathcal{F}$ a filter on $X$, and either $A$ or its complement are in $\mathcal{F}$, then $\mathcal{F}$ is ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ a filter on $X$, Suppose $\forall A \subseteq X$ where either $A$ or $X\backslash A$ are in $\mathcal{F}$, then $\mathcal{F}$ is ...
1
vote
1answer
57 views

Is my proof to show that $\mathcal{P}(A) \subseteq\mathcal{P}(B) \implies A \subseteq B$ correct? $\mathcal{P}$ refers to the power set.

Suppose $A$ and $B$ are sets, and that $x$ is an arbitrary element of $A$. The definition of the given $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ means $$\forall y[(y \in \mathcal{P}(A) \rightarrow y \...
1
vote
1answer
32 views

Cardinality of Surjective only & Injective only functions

I'm a college student just beginning to study the very basic of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
0
votes
1answer
35 views

Example of a set being subset of its power set

Could anyone please give an enlightening example of such set pls? Different from trivial one like empty set or set of empty set please? Thanks in advance
-1
votes
2answers
56 views

Determining if a power set is one to one or onto.

Let $P$ be the power set of $\{a,b,c\}$. A function $f: P \to \mathbb{Z}$; the set of integers, follows: For $A$ in $P$, $f(A)=$the number of elements in $A$. Is $f$ one-to-one? Explain. Is $f$ onto?...
0
votes
0answers
26 views

Union of an infinite set and a countable set

Let $A$ be an infinite set and $B$ be a countable set. I want to show that $|A|=|A\cup B|$. I'm aware that the following relevant definitions: An infinite set is one which has a proper subset of ...
0
votes
1answer
46 views

How can I find the sup, inf, min, and max of $\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$

$$\bigcup\left[\frac{1}{n}, 2-\frac{1}{n}\right]$$ I'm not sure how to get started with this one. When I graph the two functions I see they intersect at the point $(1,1)$, which I take to be the ...
1
vote
1answer
26 views

Determine the cardinality of $\{B\subseteq A \colon \vert B \vert \leq \kappa \}$

Let $A$ be a set. $\kappa$ a cardinal and assume that $\omega \leq \kappa \leq \vert A \vert \leq 2^{\kappa}$. Determine the cardinality of $C \colon=\{B\subseteq A \colon \vert B \vert \leq \kappa ...
1
vote
2answers
74 views

Defining “Countably Infinite”

I was reading about countably infinite sets and the definition goes as, "A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers"(Source: ...
2
votes
4answers
46 views

What are the elements of the set $\overline{X} = \{10-x \ \ |\ \ x \in \varnothing\}$?

I am having some doubts about enumerating elements of this set: $$\overline{X} = \{10-x \ \ |\ \ x \in \varnothing\}$$ Can you enumerate what elements are in there? I know this is a really ...
0
votes
1answer
35 views

Separating uncoutable number of sets preserving the sum

I was wondering -- can we make a construction that separates uncountable number of non-empty sets so that we get the same amount of elements in their sum, but our new sets are disjoint? Namely, if $$\...
0
votes
1answer
42 views

Inf and Supremum of $\{\arctan(x) \; : \; x \in \mathbb R\}$

I had trouble with this, but I think it's because there is no minimum and no maximum, correct? For the $\inf$ I got $-\frac{\pi}{2}$ and for the $\sup$ I got $\frac{\pi}{2}$ But on second thought I ...
1
vote
1answer
42 views

Let $E = \{2k$| $k\in \mathbb{N}\}$ ; $F = \{l^2$| $l\in \mathbb{N}\}$. Determine $E\cap F$, $E-F$. [closed]

I have attempted to solve these questions on my own but as I have not seen an example of this before I have no idea how to present the answer or if I am even on the right track. Any help or some type ...
1
vote
1answer
35 views

Verify sup, max, min, inf for $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$

I think I'm finally starting to understand this and wanted to check that I answered the following correctly: $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$ inf: 1 min: 4 sup: DNE max: DNE
0
votes
0answers
13 views

How do I compose a set of sets that all match on one variable element?

I have a collection of $N$ sets. Each set $n_i \in N$ is comprised of exactly $I$ elements $\{i_1, ... ,i_I\}$, where each $i_i \in J, J > I$. For any two sets $n_a$ and $n_b$, $n_a \cap n_b = \{...
0
votes
1answer
26 views

Algebra of Sets, proof verification.

Is the following simplification correct? \begin{align*} [(A \cup B \cup C)\cap(A\cup B)] - [(A \cup(B-C))\cap A] &= (A \cup B) - [(A \cap A) \cup (A \cap(B-C))]\\ &= (A \cup B) - [A \cup (A \...
0
votes
1answer
63 views

Prove that $2^\mathbb{N}$ is equinumerous to $\mathbb{R}$ [duplicate]

I was thinking using the fact that $[0,1]$ is equinumerous with $\mathbb{R}$, but i cant think of a bijection from there to $2^\mathbb{N}$.
4
votes
1answer
59 views

To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
3
votes
4answers
112 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
0
votes
1answer
27 views

Determine Intersection and difference of infinitely countable sets

I've been confused about how exactly write the answer to this question if you could help me out I would greatly appreciate it. Let $A = \{3^k$| $k\in \mathbb{N}\}$ ; $B = \{l^3$| $l\in \mathbb{N}\}$. ...
3
votes
1answer
44 views

Prove that an infinite chain of proper containments of compact sets is non empty [duplicate]

I need to prove that if $K_1\supset K_2 \supset K_3 \supset K_4 \supset \ldots$ is a chain of proper containments and each $K_{i}\subseteq \mathbb{R}^{n}$ is compact, then $\bigcap_{i=1}^{\infty} K_{i}...
2
votes
4answers
64 views

Show that the unit sphere is connected [duplicate]

I need to show that $\{(x,y,z)\in\mathbb{R}^{3}:x^2+y^2+z^2 = 1\}$ is connected. Intuitively I understand that it is path connected and, therefore, connected. However, I don't understand how I would ...
1
vote
2answers
52 views

supremum, infimum, max and min - assistance understanding the difference

I think I understand the very basic concepts of these terms, but wanted to check my understanding here. The max is the largest number in the set. The supremum is the least upper bound number ...
0
votes
2answers
32 views

Countable set of number rational, prove with $\mathbb{Z}$.

Good morning, I need to prove $ \mathbb{Q} $ is a countable set, but I prove $ \mathbb{Z} $ is a countable set, now, can I use this for proving $ \mathbb{Q} $ is countable set? I was thinking about a ...
2
votes
1answer
32 views

Understanding the solutions to questions concerning cardinalities and power sets.

Let $A = \{1, 2, 3, ... , n\}$. Find the cardinalities of the following sets: $\{(a, S) \mid a \in S, S \in P(A)\} $ $\{(S, T) \mid S \in P(A), T \in P(A), S\cap T = \emptyset \}$ ...
-2
votes
2answers
41 views

Let $A = \{1, 3, 5, 7, 9\}$ and $B=\{3, 6, 9\}$. Find each of the following: (i) $A \cup B$ (ii) $A \cap B$ (iii) $A − B$ [closed]

Let $A = \{1, 3, 5, 7, 9\}$ and $B=\{3, 6, 9\}$. Find each of the following: (i) $A \cup B$ (ii) $A \cap B$ (iii) $A − B$ I am doing a test today which I must prove this kind but honestly ...
1
vote
2answers
13 views

Trouble with a proof exercise in Set Theory regarding subsets and Power Sets

Question as posed: Let U be any set. Prove that for every $A\in\mathcal{P}(U)$ there is a unique $B\in\mathcal{P}(U)$ such that for every $C\in\mathcal{P}(U)$, $C\setminus A=C \cap B $. Proof (so far)...
0
votes
1answer
34 views

Hamel Bases: Cardinality? [duplicate]

Every vector space admits a Hamel basis by AC. That is there are maximally linear independent sets. But how to prove their cardinalities necessarily agree? ..I couldn't really find any reference.
0
votes
0answers
6 views

Set intersection with margin: terminology

I implemented an algorithm that calculates the intersection of two sets with a certain margin and returns the matched tuples: Let A, B be sets. $C = \{ (a \in A, b \in B) | lowerbound <= a - b &...
0
votes
2answers
45 views

Consider the set $S=\{-1,0, 1\}$, what is $A=\{xy: x, y\in S\}$?

Consider the set $S=\{-1,0, 1\}$ Set $A=\{xy: x, y\in S\}.$ Find all elements of $A$. Is it $A=\{-1,0\}$ or $A=\{-1,0, 1\}?$ Can you multiply $1$ by itself? Because it's $x$ and $y$....
0
votes
1answer
19 views

Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
2
votes
0answers
27 views

If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

Claim: If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$ $\coprod$ is the disjoint union of disjoint sets $A_n \subset A, \forall n \in \mathbb{N}$ Is ...
0
votes
2answers
42 views

Cardinality of the Domain vs Codomain in Surjective (non-injective) & Injective (non-surjective) functions

I'm a student in college just beginning to study the basics of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
3
votes
1answer
47 views

Is $\{\{\emptyset\}\} \subseteq \mathcal P(\{\emptyset,\{\emptyset\}\})$?

For my homework I have to determine whether $\{\{\emptyset\}\} \subseteq \mathcal P(\{\emptyset,\{\emptyset\}\})$ is true or false. I believe the answer would be true because $\{\emptyset\} \in \...
2
votes
2answers
57 views

How to show that the following function is bijective?

If we have the function $c : \mathbb{N}^2 \rightarrow \mathbb{N} : (x,y) \rightarrow 2^x \cdot (2y+1) -1 $ how to show that this function is bijective? So I thought the easiest way is to show that is ...
1
vote
2answers
34 views

An example of an infinite set with $S$ with there exists some cardinality between $S$ and $P(S)$.

I just read about Continuum Hypothesis which states that there is no set $S$ with the cardinality of $S$ is strictly larger than $\mathbb{N}$ and strictly smaller than $\mathbb{R}$. I recall that in ...
2
votes
2answers
48 views

Is $ f \circ g $ invertible in the diagram below?

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible? Thanks!
0
votes
0answers
28 views

Partition of complete boolean algebra induces partition on elements

Given a complete boolean algebra B, and two partitions W and T of B, why is it true that W induces a partition on every element of T? (And is this true more generally - does W induce a partition on ...
0
votes
1answer
32 views

If $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$

I've been thinking about the following claim: Let $A$ be a set and $|A|$ his cardinality. For every cardinal $\lambda$ with $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$. ...
-2
votes
1answer
31 views

Help understand this set theoretic equations regarding Natural Numbers

I have $\color{fuchsia}2$ problems here : $\mathbb N$- Set of all natural numbers $\Lambda=\{\lambda_n\}_n$ is a non-decreasing sequence of natural numbers such that $\lambda_1=1$ and $\lambda_{n+...
0
votes
2answers
34 views

What is the result of a natural number power the cardinality of an infinite set?

What is the result of a natural number power the cardinality of an infinite set? Is it the cardinality of the infinite set? Thank you!
3
votes
2answers
44 views

Compact Sets of $(X,d)$ with discrete metric

Let $X \neq \emptyset$. Define the discrete metric on $X$ with: $ d(x,y)=\left\{\begin{array}{ll} 1, & x \neq y \\ 0, & x=y\end{array}\right.$ (a) Ascertain the compact ...
0
votes
1answer
35 views

Cardinality of an infinite set divided by the cardinality of another infinite set (or itself) [duplicate]

Is the cardinality of an infinite set divided by the cardinality of another infinite set indeterminate? And what if it is divided by itself? Have these results been proven or are they unprovable? ...
0
votes
3answers
36 views

Prove equal cardinality between two sets?

I'm preparing for a discrete math course in September and I'm trying to study on my own this summer. I've run into a bit of trouble with a practice problem I found online and can't really figure it ...
1
vote
1answer
22 views

Notation for enumerating a set

Is there a common notation for enumerating a set? For example if $A=\{2,4,6,\ldots,n \}$ is the set of even numbers, I would like to know the notation that enumerates ordered pairs $(e,i) \in \...
0
votes
1answer
21 views

There exist partition of set $X$ due to relation $R$ and surjection $g: X\to X|_R$ and injection $h:X|_R \to Y$ such as $h \circ g=f$

$f: X\to Y$ is function. Prove: There exist partition of set $X$ due to relation $R$ on $X$ and surjection $g: X\to X|_R$ and injection $h:X|_R \to Y$ such as $h \circ g=f$