# Questions tagged [elementary-set-theory]

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, differences and complements, De Morgan's laws, Venn diagrams, relations and functions, countability and uncountability, etc. More advanced topics should use (set-theory) instead.

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### Кто может помочь доказать тождество? A ∪ В = (А \ C) ∪ (В \ А) ∪ (А ⋂ В) [closed]

A ∪ В = (А \ C) ∪ (В \ А) ∪ (А ⋂ В)
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### $f:\mathbb{R}\to\mathbb{R}$ is increasing. $\{y\in\mathbb{R}:\#f^{-1}(\{y\})\geq 2\}$ can contain at most countably many points. Why?

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. The following exercise is Exercise 24 on p.40 in Exercises 2B in this book. Exercise 24 Suppose $B\subset\mathbb{R}$...
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### Is there a proof showing that all (absolutely) normal numbers are disjunctive?

I was reading the Wikipedia pages for normal numbers and disjunctive sequences (hadn't come across either of these terms before so I'm not an expert or anything, pls be nice. Just going on a tangent ...
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### Notation: x randomly chosen with weights from set S

I have implemented $\epsilon$-greedy policy in the context of reinforcement learning in Python code: ...
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### composition relations in Set Theory

I have provided my solution and I see there are subtle differences to those provide in picture. Is my solution equivalent? Which is more rigorously correct? \begin{align} (S \circ R)^{-1} & = \...
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### Kuratowski's definition of ordered pairs: proving $(x,y) = (u,v) \iff x=u \text{ and } y=v$

I'm new to set theory and I need to prove the following: $(x,y) = (u,v) \iff x=u,y=v$ where $(x,y) = \{\{x\},\{x,y\}\}$ as in Kuratowski's definition of ordered pairs. Now, I think I got the right to ...
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### $\overline{F}:A/R\to A/R$ such that $\overline{F}([x]_R)=[F(x)]_R$ uniqueness extremely trivial im confused?

Assume that $R$ is an equivalence relation on $A$ and that $F:A\to A$. If $F$ is compatible with $R$, then there exists a unique $\overline{F}:A/R\to A/R$ such that $$\overline{F}([x]_R)=[F(x)]_R$$ ...
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