# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\aleph_{\alpha+1} \leq 2^{\aleph_\alpha}$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $S$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.$\emptyset$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...