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
33 views

Proving if $A\subseteq D$ and $B\subseteq E$ then $(D-A)\cap (E-B)\subseteq (D\cap E)-(A\cap B)$ [closed]

If $A,B,D,E$ are sets such that $A\subseteq D$ and $B\subseteq E$, how does one go about proving or disproving the following statements: $(D-A)\cap (E-B)\subseteq (D\cap E)-(A\cap B)$ $(D-B)\cup (E-...
-6
votes
2answers
82 views

About Cantor's proof of uncountability of real numbers

Here, I try to make clear what was in previous question I asked. Cantor requires one set to be countable if it exists condition that there is a bijection between natural set and the set under ...
0
votes
0answers
32 views

Method for writing set intersections/unions/complements/etc. in terms of polynomial functions? [closed]

I realize that I'm woefully uninformed regarding most math concepts, and I also realize (given the aforementioned fact) that I'm probably not using the correct format/notation, so please, be gentle. ...
1
vote
1answer
45 views

Is a monotone map between sets always continuous?

Isabelle says yes, but I am not really convinced. Given a map $f$ such that for all sets $A, B$ $A \subseteq B$ implies $f(A) \subseteq f(B)$, then is it always the case that $\bigcup_i f(S_i) = f(\...
1
vote
1answer
19 views

Return a cross product of two sets A and B such that only one entry is returned based on a condition

Lets say I have a set $A={\text{'Akshat'},\text{'John'},\text{'Mike'}}$ and a set $B={\text{'Modi'},\text{'Kerry'}}$. Set $A$ represents a set of voters while $B$ represents a set of candidates for an ...
0
votes
2answers
35 views

An elementary problem about binary relations

I am now trying to solve a research problem. I present its elementary special case so that you can participate in my research. Find binary relations $f$ and $g$ on a set $U$ such that the following ...
0
votes
3answers
42 views

How to prove the multiples of $3 $ is denumerable?

Prove that the following set is denumerable. $T$, the integer multiples of $3$. A denumerable set is that the set is equivalent to the set of natural numbers. Some of the multiples of 3 include ...
0
votes
2answers
60 views

Why does $\aleph_{\alpha+1} \leq 2^{\aleph_\alpha} $ hold true?

I proved in $\mathbb{L}$ $2^{\aleph_\alpha} \leq \aleph_{\alpha+1}$. But I don't have an idea why $\aleph_{\alpha+1} \leq 2^{\aleph_\alpha} $?
1
vote
3answers
58 views

Proving $ \bigcup_{i=1}^n A_{i} \text{ is finite.} $ by Induction.

Prove : If $A_{1},A_{2},...,A_{n} \text{ are finite sets, then } $$$ \bigcup_{i=1}^n A_{i} \text{ is finite.} $$ Proof: (I) Basis Step : $p(1)$ is true because it is true because it is finite. ...
1
vote
0answers
14 views

Arbitrary distributive law for Sets [duplicate]

How can we interchange union and intersection in $$\bigcap_{\gamma < \kappa}[\bigcup_{n<\omega}I(\gamma,n)]$$
1
vote
4answers
112 views

Trouble finding the inverse of $f(x) = x + \frac{1}{x}$ .

Let $ f: \Bbb R - \{0\} \rightarrow \Bbb R \;\text{ given by } f(x) = x + \frac{1}{x} . \text{Find} $ $f(f^{-1}(\Bbb R))$ , $\Bbb R = \text{real numbers}$. For this problem I know one needs to ...
1
vote
1answer
13 views

The least number of distinct averages

$A$ is a set of $n$ distinct real numbers where $a_i\in[0,1]$ for each $a_i$ in $A$. $F(A)$ is the set of averages of all pairs in $A$. $f_n$ is the minimal possible value of $|F(A)|$ where $|A|=n$. ...
1
vote
2answers
40 views

Box topology definition

Munkres defines the box topology as shown below. I am trying to understand how come an element of basis as defined is even subset of $\prod_{\alpha \in J}X_{\alpha}$. If we get an element of the basis,...
1
vote
2answers
28 views

Difference Between Product and Function Spaces

This confusion arose in the context of thinking about different ways to interpret stochastic processes. Say we have an index set $J$ and some other space $X$. Then is there a way to formalize ...
2
votes
0answers
30 views

Definition of “smallness” in Lurie's HTT

In Lurie's Higher Topos Theory, on the first page of the appendix, he writes If $\kappa$ is a regular cardinal, we will say that a set $S$ is $\kappa$-small if it has cardinality less than $\kappa$...
2
votes
3answers
70 views

Is $\emptyset$ considered as a powerset by itself?

For example, $X = \{\emptyset, a, \{b\}\}$. Find the power set of $X$. As far as I believe everyone understand, a power set of something means to display whatever element is within the set itself. ...
1
vote
4answers
144 views

Definition of Ordinals in Set Theory in Layman Terms

I've taken a huge interest in the mathematical concept of infinity and often been contemplating the same over years. But the fundamental concept of set theory ordinals continues to evade my ...
6
votes
2answers
547 views

Why is $\{\{1\}\}$ not equal to $\{1,\{1\}\}$?

Determine whether each of these pairs of sets are equal$$A = \{\{1\}\} \qquad \qquad B = \{1, \{1\}\}$$ I believe $A$ is equal to $B$ because all elements in $A$ are in $B$, but the answer says that ...
0
votes
0answers
25 views

A question involving symmetric differences [closed]

For any two sets $S$ and $T$, the symmetric difference $S\Delta T$ is defined as the set of all elements that belong to either $S$ or $T$ but not both, that is $S {\Delta} T = ( S\cup T) – (S \cap T)$....
4
votes
3answers
129 views

How is a set subset of its power set?

This question is from S C Kleene's Introduction to Metamathematics, page 38: If we prescribe as admissible elements of sets (a) $\varnothing$ and (b) arbitrary sets whose members are admissible ...
2
votes
3answers
23 views

Does a pairing of sets contain the constituent sets' elements?

Suppose we have two sets $$A=\{1,2,3\} \qquad \qquad B=\{4,5,6\}$$ We now define the set $C=\{A,B\}$. Are the elements of $A$ and $B$ now elements of $C$? Or are $A$ and $B$ the elements of the set? ...
14
votes
4answers
621 views

Uncountability of increasing functions on N

I believe I have made a reasonable attempt to answer the following question. I would like a confirmation of my proof to be correct, or help as to why it is incorrect. Question: Let $f : \mathbb{N} \...
0
votes
1answer
34 views

Express that a set is finite using symbols

Two related questions: What's the most elegant way to express that the set $S$ is finite using logical symbols? Obviously this will depend to some extent on what you allow yourself to quantify, so ...
1
vote
1answer
21 views

If $P(A)=\{\varnothing,\{\varnothing\}\}$ then find $P(P(A)-A)$

If $P(A)=\{\varnothing,\{\varnothing\}\}$ then find $P(P(A)-A)$. note that $P$ is the set of subsets of $A$. My solution:I find the answer same as $P(A)$. Am I right? I asked this because I had a ...
1
vote
1answer
56 views

Show that a set with an uncountable subset is itself uncountable.

Let $A = P \cup Q$, where $P, Q$ are disjoint [1] and $P \ne \emptyset$ is countable and $Q \ne \emptyset$ is uncountable. Then $Q \subset A$ [2]. Show that $A$ is uncountable. Proof (by ...
0
votes
2answers
27 views

Relating taking the power set to logical operations

I'm an undergraduate math major reviewing "Mathematical Proofs, A Transition to Advanced Mathematics" and specifically the first two chapters on sets and logic. I'm trying to find ways to write set ...
1
vote
2answers
53 views

Is $\{a, b\}$ $\subset$ $\{a, b, c\}$ the same thing as $\{\{a, b\}\}$ $\subset$ $\{a, b, c\}$?

It's clear to me that $\{a, b\}$ $\subset$ $\{a, b, c\} = S$. But I don't see the element $\{\{a, b\}\}$, as a whole, inside $S$.
3
votes
2answers
31 views

Quantification = statement about an open sentence?

The book I'm reading is talking about quantification being a method to convert open sentences into statements. From what I can see this method boils down to making a statement about the solution set ...
2
votes
2answers
38 views

Inductive function

I want to understand what is meant by an inductive set. I've found it to be defined as: if z is in a set K then $$z\cup \{z\} $$ is in the set. How is this possible since the reunion is between sets ...
1
vote
1answer
42 views

Infinite sum of nonnegative USC functions

Let $\{f_n\}$ be sequence of real nonnegative functions on $\mathbb{R}^1$, and consider the following statement: If each $f_n$ is upper semicontinuous (USC), then $\sum \limits_{1}^{\infty} f_n$ is ...
1
vote
1answer
56 views

What does multiply mean in a set

I have seen this question today. $[A \cdot(B-C)]\cap[(A \cup C)\cdot(C-B)]$ 1.$(A-B) \cdot (A-C)$ 2.$(A \cdot B)\Delta (A\cdot C)$ 3.$(A \cup B)\cdot (A \cup C)$ 4.$\emptyset$ ...
1
vote
1answer
43 views

Is “closedness” a proper word?

In one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward to ...
0
votes
1answer
21 views

Is $I \cap (\bigcup_{j=0}^{\infty} I_j)$ a half open interval?

A half open interval is a set of form $\emptyset$ or $[a,b[$ where $a < b$ If $I$ is a half open interval and $I_j, j=0,1,...$ is a sequence of half open intervals, is $I \cap (\bigcup_{j=0}^{\...
0
votes
0answers
32 views

How can I represent this set?

I have a set of variables: $X := \{x_1, \ldots, x_n\}$ Let the range for each variable $x_i$ be $R_i$. How can I represent the set $\{(x_i, v_i), \forall x_i \in X, \forall v_i \in R_i \}$ in terms ...
1
vote
6answers
64 views

Definition of Relation of a Set

The standard definition of a relation of an arbitrary set A is a subset of the set product of A, AxA. Is it okay to define relation R to be a subset of the set product AxA such that R has at least ...
0
votes
1answer
21 views

Intersecting a set with an arbitrary union

I want to verify that my proof is correct for the following fact $\bigcup_{\alpha \in \mathbb{J}} (U_\alpha \cap Y) = (\bigcup_{\alpha \in \mathbb{J}} U_\alpha)\cap Y$. let $x \in \bigcup_{\alpha \...
2
votes
1answer
24 views

Transitive Property of Proper Inclusion

$ Theorem:$ If $ A \subset B $ and $B \subset C$, then $A \subset C$. Here $X \subset Y$ is defined as $ X\subseteq Y$ and $X\neq Y$. I can prove the case of improper inclusion using the ...
-4
votes
5answers
102 views

About Cantor's proof of uncountability or real numbers

The proof is using reductio ad absurdum , i.e. contradiction. Start with that there is a sequence of all real numbers (in some interval) and then it is shown that there exists number that is different ...
0
votes
1answer
59 views

Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
0
votes
0answers
13 views

Explaining the proof that the set of integer combinations of a,b is equal to the set of integer multiples of the GCD of a,b

I'm trying to understand how this proof works, but the website is using a different notation than my book and it's not making any sense to me. here is the proof on ProofWiki I know that the ...
0
votes
1answer
20 views

Isn't $\bigcap_{i=1}^5 A_i=A_5$ where $A_i=\{x \mid x \in \mathbb{N},0 \le x \le \frac{1}{n}\}$

If $A_i=\{x \mid x \in \mathbb{N},0 \le x \le \frac{1}{n}\}$ where $n \in \mathbb{N}$. which one is correct?(Note that $0 \notin \mathbb{N}$) 1.$\bigcup_{i=1}^n A_i=A_1$ 2.$\bigcap_{i=1}^5 ...
0
votes
1answer
27 views

Cartesian product of a family of sets by halmos [duplicate]

can someone please give me an example for this sentence, (i don't want a formal definition) thank's.
3
votes
1answer
28 views

Is a relation between A and B the same as a mapping from elements of A to subsets of B?

The way I always saw it was that a relation is a subset of $A \times B$, or a collection of ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Is there any meaningful distinction between the two ...
1
vote
4answers
58 views

prove $\bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty B_n$

prove$\bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty B_n$ if $A_i$ is a arbitrary set and $B_1=A_1$ and $B_n=A_n-\bigcup_{i=1}^{n-1} A_i$ for $n\ge 2$. I have no ideas for solving it there was another ...
0
votes
1answer
38 views

Proof of Ultrafilter lemma with two propositions and Zorn lemma

I would like to prove the following: Let $X$ be any set, then every filter $\mathcal{F}$ on $X$ is contained in an ultrafilter $F$ Using two propositions and Zorn Lemma. I am required to come ...
0
votes
3answers
33 views

Set theory confusion

Given a universal set $\{ 1,2,3,4,5,6,7,8,9 \} $, $A$ is defined as ... $A = \{ x : (x-1)(x-6) \lt 0 \}$ So what are the elements in $A$ , I'm a little confused here . The $x$ values is $= 1$ or ...
0
votes
1answer
26 views

formula for defining terms in a finite set

Suppose there's a finite set, $S$ of terms in $\mathbb{R}$ which have the property $P(x)$. Suppose we know how to define the maximum value of the set by the relation, $max(x)$. We also have the ...
0
votes
3answers
35 views

How to write half-open intervals as disjoint ones

I have a collection of half open intervals $(I_j, j \in \mathbb N)$ and I want to get a new collection $(J_j, j \in \mathbb N)$ out of the $I_j$ such that for $i \neq j$ $J_j \cap J_i = \emptyset$ and ...
0
votes
2answers
34 views

Proving the Inclusion-Exclusion Formula?

I've been given the following problem: ...
0
votes
1answer
18 views

Prove there is a $g:B\rightarrow\mathbb{R}$ s.t. $f(g(b))=b$ for each $b\in B$.

Suppose $f:\mathbb{R}\rightarrow B$ is surjective, where $B$ is finite. Prove there is a $g:B\rightarrow\mathbb{R}$ s.t. $f(g(b))=b$ for each $b\in B$. I'm given a hint to use induction on the ...