#settheory — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #settheory, aggregated by home.social.
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Alright, future engineers!
**Power Set:** The set of all subsets of a set, including empty & the set itself.
Ex: S={A,B}, P(S) = {{}, {A}, {B}, {A,B}}.
Pro-Tip: A set with 'n' elements has `2^n` subsets. Essential for understanding relations!
#SetTheory #DiscreteMath #STEM #StudyNotes -
New blog post “‘The road to ε₀: Shortlex order also orders sequences of numbers”
In finite Nim, the nim-heaps are just ordinary finite numbers. In infinite Nim, they can be any finite _sequences_ of finite numbers, and if we use shortlex to order them, the game is still well-founded: It must always end eventually.
The previously difficult-seeming number \( ω^ω \) is just revealed as the type of finite sequences of finite numbers, not so difficult at all!
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Alright, future engineers!
**Set Union (A U B):** The set containing all elements that are in set A, or in set B, or in both.
Ex: A={1,2}, B={2,3} => A U B = {1,2,3}.
Pro-Tip: Think 'OR' – an element is in the union if it's in A *OR* in B. Visualize with Venn Diagrams! -
Alright, future engineers!
**Cartesian Product:** A set of all possible ordered pairs where the first element is from the first set & the second from the second.
Ex: A={1,2}, B={a,b} -> AxB = {(1,a), (1,b), (2,a), (2,b)}
Pro-Tip: `|AxB| = |A| * |B|`. Essential for combining domains!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Power Set:** The set of all possible subsets of a given set.
Ex: For A={1,2}, P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If a set has 'n' elements, its power set will always have 2^n elements. Count 'em!
#SetTheory #DiscreteMath #STEM #StudyNotes -
🎩 Behold, the mathematical wizardry of Joel David Hamkins, who has boldly declared that sets of natural numbers have a rich #lattice structure! 🤔 Truly #groundbreaking, except for the tiny detail that everyone with a basic understanding of set theory already knows this. Bravo, Professor Hamkins, for turning the obvious into a revelation! 🎉
https://jdh.hamkins.org/the-lattice-of-sets-of-natural-numbers-is-rich/ #mathematics #settheory #academiccommunity #naturalnumbers #HackerNews #ngated -
Alright, future engineers!
**Power Set (P(S)):** The set of all possible subsets of a given set S, including the empty set & S itself.
Ex: If S={1,2}, P(S) = {{}, {1}, {2}, {1,2}}. Size is 2^n.
Pro-Tip: Every element in P(S) is itself a set!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Power Set:** The set of *all* subsets of a given set, including the empty set & the set itself.
Ex: If A={1,2}, P(A)={{},{1},{2},{1,2}}.
Pro-Tip: If a set has 'n' elements, its power set has 2^n elements!
#SetTheory #DiscreteMath #STEM #StudyNotes -
New blog post “‘The road to ε₀: Coin-moving games with no coins”
I continue to approach ε₀ in small stages. We take the previous model of \( ω^ω \), an infinite-dimensional array of cells, and reinterpret it as a much simpler model about finite sequences of numbers.
I also get to write the expression \[ ω^{ω^{ω^{ω^⋰}}}. \]
https://blog.plover.com/math/ordinals/04-coordinates.html
#math #blog #setTheory #countableOrdinals #universeOfDiscourse
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Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set.
Ex: For `A = {1, 2}`, `P(A) = { {}, {1}, {2}, {1, 2} }`.
Pro-Tip: If a set has `n` elements, its power set has `2^n` elements!
#SetTheory #DiscreteMath #STEM #StudyNotes -
MVspa_ Stochastic processes
Стохастические процессы, Випадкові процеси
#Stochastic processes
#Стохастическиепроцессы, #Випадковіпроцеси
t.me/scilib_yura15cbx/230MVsa Statistics and applications
Статистика та додатки, Статистика и приложения
#Statistics
t.me/scilib_yura15cbx/229EM_ Mechanics of elastic materials
Механіка пружних матеріалів, Механика упругих материалов
t.me/scilib_yura15cbx/228MPd_ Dynamical systems
Динамические системы, Динамічні системи
#Dynamical systems
t.me/scilib_yura15cbx/227MNw_ Wavelets, signal processing
Вейвлети, обробка сигналів, Вейвлеты, обработка сигналов
#Wavelets, #signal processing
t.me/scilib_yura15cbx/226MNs_ Simulations of physics
Физические симуляции
t.me/scilib_yura15cbx/225MNl_ Numerical linear algebra
Числова лінійна алгебра, Численная линейная алгебра
t.me/scilib_yura15cbx/224MNf_ Finite elements
Конечные элементы, Скінченні елементи
t.me/scilib_yura15cbx/223MNd_ Numerical calculus
t.me/scilib_yura15cbx/222MDgt_ General topology
#topology
Общая топология, Загальна топологія
t.me/scilib_yura15cbx/221MDdg_ Differential geometry
Диференціальна геометрія, Дифференциальная геометрия
t.me/scilib_yura15cbx/220MDat_ Algebraic and differential topology
Алгебраическая и дифференциальная топология, Алгебраїчна та диференціальна топологія
t.me/scilib_yura15cbx/219MCv_ Variational calculus
Варіаційне числення, Вариационное исчисление
t.me/scilib_yura15cbx/218MCta_ Tensor calculus, forms
Тензорное исчисление, формы, Тензорне числення, форми
t.me/scilib_yura15cbx/217MCsf_ Special functions
Спеціальні функції, Специальные функции
t.me/scilib_yura15cbx/216MCf_ Functional analysis
Функциональный анализ, Функціональний аналіз
t.me/scilib_yura15cbx/215MCet_ Elementary calculus textbooks
Підручники з елементарного числення, Учебники по элементарному исчислению
t.me/scilib_yura15cbx/214MCde_ Differential equations
Дифференциальные уравнения, Диференціальні рівняння
#Differential equations
t.me/scilib_yura15cbx/213MCcf_ Continued fractions
Суцільні дроби, Непрерывные дроби
t.me/scilib_yura15cbx/212MCc_ Complex variable
Комплексная переменная, Комплексна змінна
t.me/scilib_yura15cbx/211MCat_ Advanced calculus
t.me/scilib_yura15cbx/210Asymptotics, perturbations
Асимптотика, збурення,
Асимптотика, возмущения
#Asymptotics, #perturbations
#Асимптотика, #збурення,
#Асимптотика, #возмущения
t.me/scilib_yura15cbx/209Group theory, Теория групп, Теорія груп
#Group theory, #Теориягрупп, #Теоріягруп
t.me/scilib_yura15cbx/208Algebra textbooks,
Підручники з алгебри,
Учебники по алгебре
t.me/scilib_yura15cbx/207Set theory,
Теория множеств, Теорія множин
#Set theory,
#Теория множеств,
t.me/scilib_yura15cbx/206Representation theory
Теорія репрезентації, Теория представлений
t.me/scilib_yura15cbx/205Quantum groups
#Quantum groups
Квантовые группы, Квантові групи
t.me/scilib_yura15cbx/204Mathematical logic
#Mathematical logic
Математична логіка, Математическая логика
t.me/scilib_yura15cbx/203Linear algebra
#Linear algebra
Линейная алгебра, Лінійна алгебра
t.me/scilib_yura15cbx/202Homology, Гомологія, Гомология
#Homology, #Гомологія, #Гомология
t.me/scilib_yura15cbx/201Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Algebraic geometry
#Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Finite groups
Скінченні групи, Конечные группы
t.me/scilib_yura15cbx/199Differential algebra
#Differential algebra
Дифференциальная алгебра, Диференціальна алгебра
t.me/scilib_yura15cbx/198Category theory
Теорія категорій, Теория категорий -
MVspa_ Stochastic processes
Стохастические процессы, Випадкові процеси
#Stochastic processes
#Стохастическиепроцессы, #Випадковіпроцеси
t.me/scilib_yura15cbx/230MVsa Statistics and applications
Статистика та додатки, Статистика и приложения
#Statistics
t.me/scilib_yura15cbx/229EM_ Mechanics of elastic materials
Механіка пружних матеріалів, Механика упругих материалов
t.me/scilib_yura15cbx/228MPd_ Dynamical systems
Динамические системы, Динамічні системи
#Dynamical systems
t.me/scilib_yura15cbx/227MNw_ Wavelets, signal processing
Вейвлети, обробка сигналів, Вейвлеты, обработка сигналов
#Wavelets, #signal processing
t.me/scilib_yura15cbx/226MNs_ Simulations of physics
Физические симуляции
t.me/scilib_yura15cbx/225MNl_ Numerical linear algebra
Числова лінійна алгебра, Численная линейная алгебра
t.me/scilib_yura15cbx/224MNf_ Finite elements
Конечные элементы, Скінченні елементи
t.me/scilib_yura15cbx/223MNd_ Numerical calculus
t.me/scilib_yura15cbx/222MDgt_ General topology
#topology
Общая топология, Загальна топологія
t.me/scilib_yura15cbx/221MDdg_ Differential geometry
Диференціальна геометрія, Дифференциальная геометрия
t.me/scilib_yura15cbx/220MDat_ Algebraic and differential topology
Алгебраическая и дифференциальная топология, Алгебраїчна та диференціальна топологія
t.me/scilib_yura15cbx/219MCv_ Variational calculus
Варіаційне числення, Вариационное исчисление
t.me/scilib_yura15cbx/218MCta_ Tensor calculus, forms
Тензорное исчисление, формы, Тензорне числення, форми
t.me/scilib_yura15cbx/217MCsf_ Special functions
Спеціальні функції, Специальные функции
t.me/scilib_yura15cbx/216MCf_ Functional analysis
Функциональный анализ, Функціональний аналіз
t.me/scilib_yura15cbx/215MCet_ Elementary calculus textbooks
Підручники з елементарного числення, Учебники по элементарному исчислению
t.me/scilib_yura15cbx/214MCde_ Differential equations
Дифференциальные уравнения, Диференціальні рівняння
#Differential equations
t.me/scilib_yura15cbx/213MCcf_ Continued fractions
Суцільні дроби, Непрерывные дроби
t.me/scilib_yura15cbx/212MCc_ Complex variable
Комплексная переменная, Комплексна змінна
t.me/scilib_yura15cbx/211MCat_ Advanced calculus
t.me/scilib_yura15cbx/210Asymptotics, perturbations
Асимптотика, збурення,
Асимптотика, возмущения
#Asymptotics, #perturbations
#Асимптотика, #збурення,
#Асимптотика, #возмущения
t.me/scilib_yura15cbx/209Group theory, Теория групп, Теорія груп
#Group theory, #Теориягрупп, #Теоріягруп
t.me/scilib_yura15cbx/208Algebra textbooks,
Підручники з алгебри,
Учебники по алгебре
t.me/scilib_yura15cbx/207Set theory,
Теория множеств, Теорія множин
#Set theory,
#Теория множеств,
t.me/scilib_yura15cbx/206Representation theory
Теорія репрезентації, Теория представлений
t.me/scilib_yura15cbx/205Quantum groups
#Quantum groups
Квантовые группы, Квантові групи
t.me/scilib_yura15cbx/204Mathematical logic
#Mathematical logic
Математична логіка, Математическая логика
t.me/scilib_yura15cbx/203Linear algebra
#Linear algebra
Линейная алгебра, Лінійна алгебра
t.me/scilib_yura15cbx/202Homology, Гомологія, Гомология
#Homology, #Гомологія, #Гомология
t.me/scilib_yura15cbx/201Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Algebraic geometry
#Algebraic geometry
Алгебраическая геометрия, Алгебраїчна геометрія
t.me/scilib_yura15cbx/200Finite groups
Скінченні групи, Конечные группы
t.me/scilib_yura15cbx/199Differential algebra
#Differential algebra
Дифференциальная алгебра, Диференціальна алгебра
t.me/scilib_yura15cbx/198Category theory
Теорія категорій, Теория категорий -
New blog post “‘The road to ε₀: Infinite Nim as a coin-moving game”
I continue to approach ε₀ in small stages. Previously, nim-heaps got us to ω, and nim-heaps plus special green tokens to ω². In this article we reformulate Nim as a game about moving coins around on an array of squares, and this takes us to \( ω^ω \).
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Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set, including the empty set & the set itself.
Ex: For A={1,2}, P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If a set has 'n' elements, its Power Set always has 2^n elements!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, engineers!
**Power Set:** The set of *all subsets* of a given set. (It includes the empty set and the set itself!)
Ex: For S={1,2}, P(S) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: A set with 'n' elements always yields 2^n subsets!
#DiscreteMath #SetTheory #STEM #StudyNotes -
New blog post “The road to \( \epsilon_0 \): Nim always ends, even with infinite ordinals
A bit of a sidetrack, but too amusing to skip: even with nim-heaps that correspond to absurdly large infinite ordinals, the game itself is still finite and must end after a finite number of moves.
Also, programmers who can't tell you when they will give you a project estimate, but can tell you when they will be able to tell you.
https://blog.plover.com/math/ordinals/02-wellfoundedness.html
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Alright, future engineers!
**Power Set:** The set of ALL possible subsets of a given set.
Ex: A={1,2}. P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If |A|=n, then |P(A)| = 2^n. Essential for understanding relationships & logic!
#SetTheory #DiscreteMath #STEM #StudyNotes -
New blog post “The road to \( \epsilon_0 \): ordinals as nim-heaps”.
I'm starting a series of articles about understanding countable ordinal numbers, with the goal of getting to an understanding of the one called \( \epsilon_0 \). It'll be a long and somewhat winding path. This one discusses how ordinals, even infinite ordinals, can be interpreted as positions in the game of Nim.
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Alright, future engineers!
**Cartesian Product:** Ordered pairs (a,b) where 'a' is from A, 'b' from B.
Ex: A={1,2}, B={x,y} -> AxB = {(1,x), (1,y), (2,x), (2,y)}.
Pro-Tip: Size = |A| * |B|! Fundamental for relations & database joins.
#DiscreteMath #SetTheory #STEM #StudyNotes -
Alright, future engineers!
**Cartesian Product:** Creates a set of all possible *ordered pairs* between elements of two sets.
Ex: If A={1,2} & B={x,y}, then A x B = {(1,x), (1,y), (2,x), (2,y)}.
Pro-Tip: Think 'coordinates'! The order of elements in the pair always matters.
#SetTheory #DiscreteMath #STEM #StudyNotes -
**Inclusion-Exclusion Principle:** Counts elements in a union of sets by adding individual sizes, then subtracting overlaps.
Ex: `|A U B| = |A| + |B| - |A n B|`
Pro-Tip: Essential for complex counting problems, prevents double-counting! -
New blog post “Starting to understand epsilon-zero”.
A long time ago I said that \( ω^ω \) was the place where the countable ordinals start to get scary, and then about 18 months ago I realized I could think about it in a different way that made it not scary at all.
Then I realized that by taking that insight a little farther I could really get my head around \( \epsilon_0 \) for the first time.
This isn't either of those articles, it's an introduction that explains infinite ordinals and introduces what \( ω^ω \) and \( \epsilon_0 \) are, so that I can move on to those other two articles without explaining everything from first principles.
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Alright, future engineers!
**Power Set (P(A)):** The set of all subsets of A, including the empty set & A itself.
Ex: For `A={1,2}`, `P(A) = {{}, {1}, {2}, {1,2}}`
Pro-Tip: If A has `n` elements, `P(A)` always has `2^n` subsets! Handy for counting.
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Power Set:** The set of all possible subsets of a given set, including the empty set.
Ex: If A = {a, b}, P(A) = {{}, {a}, {b}, {a, b}}. Number of subsets = `2^|A|`.
Pro-Tip: Useful for enumerating all configurations or choices from a set!
#DiscreteMath #SetTheory #STEM #StudyNotes -
Alright, future engineers!
**Set Union (A U B):** All elements in set A, set B, or both.
Ex: If A={1,2}, B={2,3}, then A U B = {1,2,3}.
Pro-Tip: Think OR! An element is in A U B if it's in A OR B.
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
**Set Cardinality** is the number of distinct elements in a set.
Ex: For `A = {apple, banana, apple}`, `|A| = 2`.
Pro-Tip: Don't count duplicates! Each element is counted only once.
#DiscreteMath #SetTheory #STEM #StudyNotes