Questions tagged [elementary-set-theory]
For elementary questions on set theory. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations and countability.
28,753
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Proof when sets converge to the Cantor set and are decreasing, does this hold generally?
I have a question regarding a proof, first I will give the statement and the proof:
Define $C_0=[0,1]$, define
$C_n=\frac{C_{n-1}}{3}\cup(\frac{2}{3}+\frac{C_{n-1}}{3})$. Then
$C_{n+1}\subset C_{n}$.
...
- 1,354
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Can the closure operator arising from a symmetric, anti-reflexive relation be trivial without the relation being maximal?
This question originates from my attempt to study certain closure operators that arise from symmetric, anti-reflexive relations. To be precise, consider a relation $\perp$ on a set $X$ that is ...
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Measure of difference of countable unions
Let $(X,\mathcal{M},\mu)$ a measurable space, $\mu$ positive measure.
For all $n\in \mathbb{N}$, let $A_n,B_n\in \mathcal{M}$ s.t. $A_n\subseteq B_n$ and $\mu(B_n\setminus A_n)=0$.
I want to deduce $$\...
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To check if a relation is a poset or a total order or an equivalence relation.
$F$ is the set of functions defined from $\mathbb{N}\cup \{0\}$ to the set $\mathbb{R}_+$ of positive real numbers. $R$ is a relation on $F,$ defined by
$$R=\{(f,g) \mid \exists x_0\in\mathbb{N},\...
- 4,520
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To check if a given relation on a set is poset.
Given a relation $R$ on sets defined by
$R=\{(A,B):A∩B=B\}$. I have to check whether it is poset or not?
My attempt:
I assumed that $A∩B=B$ gives the set which is on the RHS of intersection.
So, ...
- 4,520
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A little trick with infinite union and complement
Let $X$ be a set, I want to show that
$$A_n\subseteq B_n\subseteq X, \hspace{5pt} \forall n\in \mathbb{N} \Longrightarrow \bigcup_{n\in \mathbb{N}}B_n \setminus \bigcup_{n\in \mathbb{N}}A_n=\bigcup_{n\...
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Sets with infinite nestings and ZFC [closed]
The set x = {$x_0, x_1, x_2$, …} with … $\in x_2 \in x_1 \in x_0$ is not a set in ZFC due to the regularity axiom because x $\cap x_n$ = {$x_{n+1}$}.
But what is with the set x‘ = {… $\in$ $x_2$ $\in$ ...
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2
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Set Theory question about the Empty Set [closed]
Not interested in answers specific to certain set theories (ZFC, etc.), just general concepts.
I have a pretty good idea what the empty set is for, and that basically everything in set theory follows ...
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What are the parts of a set operation result called?
Let's take some set operation, like union or intersection.
The term "set operation" refers to the action to be performed. Is there a name for the result, as an analogue for "sum" ...
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3
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Show that $\{(x,y)\in\mathbb{R}^2\mid x>0, 0<y<x^2\}$ is not a star domain
Show that $M:=\{(x,y\in\mathbb{R}^2\mid x>0, 0<y<x^2\}$ is not a star domain.
If I draw a picture, then it's easy to verify that due to the slope of $x^2$ we can always find a point $(a,b)\...
- 4,712
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Union of a finite or countable number of uncountable sets
Prove that the union of a finite or countable number of sets
each of power $c$ (continuum power) is itself of power $c$.
To prove this I would start considering two sets $A$ and $B$ of power $c$. I ...
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Suppose two sets $X,Y$ are such that $|X| < |Y|$ and $X$ is infinite. Prove that $Y$ must also be an infinite set.
Suppose two sets $X,Y$ are such that $|X| < |Y|$ and $X$ is infinite. Prove that $Y$ must also be an infinite set.
Since $X$ has a cardinality strictly lesser than that of $Y$, $\exists$ $f : X \to ...
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Does this set have an $\epsilon$-minimal element?
This set has two elements, one of which is the empty set. The other element is an infinite descending chain. But this set seems to have an $\epsilon$-minimal element (and so does its nonempty element) ...
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Finding infimum and supremum of a set of cyclic sums
I am trying to find the supremum and infimum of the following set:
$$
\left\{ \frac{a_1}{a_1 + a_2 + a_3} + \frac{a_2}{a_2 + a_3 + a_1} + \cdots + \frac{a_{n-2}}{a_{n-2} + a_{n-1} + a_n} + \frac{a_{n-...
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Summation of a subset of numbers from given intervals.
There are $100$ integers on the board: the first number is between $0$ and $99$ (inclusive), the second is between $-1$ and $98$, ...., the $100$th is between $-99$ and $0$. Prove that the sum of some ...
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