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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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
111 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
63 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
46 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
12 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
33 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
26 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
37 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
45 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
55 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
32 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
0answers
41 views

Geometric interpretation of Cantor's infinite list of real numbers

Cantor imagined the list of real numbers and demonstrated by diagonalization that the list can never be complete, but I wonder if such a diagonalization is even possible. If we try to build his list ...
2
votes
2answers
47 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
25 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
29 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
43 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
33 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
35 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
20 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 $h \circ g=f$
0
votes
1answer
37 views

Proof of a Surjective Function

I've run into a question in my textbook and I'm not sure if I understand fully the answer from the solution manual. Here is the question: Problem: Suppose that $f: A \rightarrow B$ is any function. ...
2
votes
3answers
44 views

Is this a bijective function for $f:(0,1) \rightarrow (-2,5)$?

$f:(0,1) \rightarrow (-2,5)$ I'm basically trying to prove the two intervals above have the same cardinality by finding a bijective function. I'm not sure I did it properly but the function I found ...
1
vote
2answers
46 views

Given $C \subset A \subset X$, why is that $C$ is closed in $X$ if $A$ is closed, $C$ is open in $X$ if $A$ is open?

I want to understand a result discussed here : Subspace of a normal space Let $(X, \mathfrak{T})$ be a topological space. Given $C \subset A \subset X$, let $C$ be a closed set in $A$, then claim ...
0
votes
2answers
50 views

Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$

Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$ I am to assume the intervals have the same cardinality. I honestly don't even know how to begin with this. Can you provide me ...
1
vote
1answer
26 views

Finding set of functions

$ f\left(u,v\right)=u^{2}+3v^{2} $ $g\left(x,y\right)=\begin{pmatrix} e^{x}cosy \\ e^{x}siny \end{pmatrix} $ How do I determine sets of $f\left(\mathbb R ^{2} \right)$ and set of $g\left(\mathbb ...
-2
votes
0answers
17 views

Existence Theorem of Natural Recursion & Mutual Recursion

$f: A -> R$ $f(a)=$ $1. basecase$ $2. g(...h(first loa) f(rest loa))$ Exist a function $F: N-> R$ Q: Is there any theorem that says the existence of this relation? More complicated version ...
3
votes
2answers
36 views

Notation for conditional set complement?

As far as I know, given $U=\{1,2,3,4,5,6\},A=\{1,2,3\}$ the notation for its set complement is $A^C = \{4,5,6\}$ Is there any sort of notation for a conditional set complement? For example, lets say ...
0
votes
2answers
23 views

Description of preimage of $g$ defined by $g|_A = f|_A, g(S \setminus A) = \{x\}$

$S$ is a given directed set. $X$ is a first countable topological space. There is a function $f : S\rightarrow X$. With help of this $f$ , a new function $g:S\rightarrow X$ is being defined like the ...
0
votes
1answer
19 views

What is $(A_1 \times … \times A_n) \cup(B_1 \times … \times B_n)=?$ ,$A_i$'s are intervals

What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively. What I mean is can $(A_1 \times ... \...
2
votes
1answer
40 views

Are there any constructive axioms which disprove the continuum hypothesis?

I understand that the Continuum hypothesis is independent of ZFC, so that we may comfortably add either the continuum hypothesis or its negation to ZFC without creating any paradoxes (unless ZFC had ...
1
vote
2answers
60 views

How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
1
vote
2answers
42 views

How does the infinite union work in $\sigma$-algebra?

Reading a probability theory book, and it says that if we have a sample space $\Omega$, then some class $F$ of subsets of $\Omega$ makes a $\sigma$-algebra and has the following properties: 1) $\...
2
votes
0answers
27 views

Find the sets for $f:R \rightarrow R$ given by $f(x) = \left|x\right|$

Find the sets for $f:R \rightarrow R$ given by $f(x) = \left|x\right|$. Let $S= [0,4]$ and $T = [-3,0]$. Find the sets: $f(S)$, $f(T)$, $f(S) \cap f(T)$, and $f(S \cap T)$. Is $f(S) \cap f(T) = f(...
3
votes
0answers
42 views

My proof that $f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2)$

I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ...
0
votes
0answers
62 views

When is it true that $V \subseteq \overline V \subseteq U$ will hold for open sets?

Let $(X, \mathfrak{T})$ be a topological space Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$ Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is ...
2
votes
2answers
138 views

Is the class of imaginable objects which cannot exist a worthwhile thing to talk about?

Clearly, in mathematics, there are certain objects which we can imagine, yet which have no real meaning, or do not exist. The real number whose square is negative, the quantity represented by ${1 \...
4
votes
2answers
45 views

My proof that $f[f^{-1}(D)] \subseteq D.$

I've just started studying formal proof and set theory, so it'll be really cool if someone can check out my proof for a pretty basic set theory problem. It'll be great if you can tell me if my proof ...
0
votes
1answer
29 views

Transitive closure of $p=\{(1,3),(2,1),(3,2),(4,1)\}$

What is the transitive closure of the relation $p$? I thought it would just be $t=p \cup p^2$. But in the solution I have, there is also $p^3$. Why is this so? What I showed is already the smallest ...
0
votes
1answer
52 views

Difficulty in choosing correct answer among the options.

1) The Cantor set, a subset of the real numbers: A. is not compact. B. is not contained in an interval. C. does not contain a non-trivial interval. D. does not have uncountably ...