#noindex
[[집합,set]] 이론

Up:
 [[수학,math]]
 [[집합과_확률,set_and_probability]]
 [[집합,set]] [[이론,theory]]

Sub:
 // [[분할,partition]]:
 [[집합의_분할,set_partition]]
 [[자연수의_분할,integer_partition]]

 // 이 둘은 대비됨
 소박한집합론 naive_set_theory https://proofwiki.org/wiki/Definition:Naive_Set_Theory [[WpEn:Naive_set_theory]] { '''naïve set theory / naive set theory''' Rel. WpKo:부랄리포르티_역설 } // "naive set theory" Ggl:"naive set theory"
 공리적집합론 axiomatic_set_theory https://proofwiki.org/wiki/Definition:Axiomatic_Set_Theory { mkl [[공리,axiom]] }

 // 이 둘도 대비됨.
 structural_set_theory https://ncatlab.org/nlab/show/structural+set+theory ..... 구조적집합론? 구조집합론? { mkl [[구조,structure]] }
  zeroth-order_set_theory https://ncatlab.org/nlab/show/zeroth-order+set+theory
  first-order_set_theory https://ncatlab.org/nlab/show/first-order+set+theory
  higher-order_set_theory https://ncatlab.org/nlab/show/higher-order+set+theory
 material_set_theory
  AKA membership-based set theory
  ex. ZFC
  sub - 순집합? 순수집합? [[,pure_set]] https://ncatlab.org/nlab/show/pure+set
  https://ncatlab.org/nlab/show/material+set+theory
 see also https://ncatlab.org/nlab/show/material-structural+adjunction

 descriptive_set_theory - 작성중
  https://mathworld.wolfram.com/DescriptiveSetTheory.html

 pure_set_theory pure set theory "is a system of set theory in which all elements of sets are themselves sets."(pw) https://proofwiki.org/wiki/Definition:Pure_Set_Theory

 // 아래 MKL: axiomatic_set_theory 

 Z집합론 Zermelo_set_theory ? or Z_set_theory ?
  Zermelo set theory
  [[WpEn:Zermelo_set_theory]] = https://en.wikipedia.org/wiki/Zermelo_set_theory
  https://plato.stanford.edu/entries/zermelo-set-theory/
  "Zermelo set theory"
  Ggl:"Zermelo set theory"
  // [[Ernst_Zermelo]] { Ernst Zermelo WpEn:Ernst_Zermelo }
 ZF집합론 ZF_set_theory ? or Zermelo-Fraenkel_set_theory ?
  Zermelo-Fraenkel set theory
  [[https://terms.naver.com/entry.naver?docId=4125139&cid=60207&categoryId=60207 수학백과 ZF 집합론]]
  Libre:체르멜로-프렝켈_집합론
  [[WpKo:체르멜로-프렝켈_집합론]] = https://ko.wikipedia.org/wiki/체르멜로-프렝켈_집합론
  [[WpSp:Zermelo–Fraenkel_set_theory]] = https://simple.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
  [[WpEn:Zermelo–Fraenkel_set_theory]] = https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
  [[WpJa:ツェルメロ＝フレンケル集合論]] = https://ja.wikipedia.org/wiki/ツェルメロ＝フレンケル集合論
  https://plato.stanford.edu/entries/set-theory/ZF.html
  "Zermelo-Fraenkel set theory"
  Ggl:"Zermelo-Fraenkel set theory"
  // [[Abraham_Fraenkel]] { Abraham Fraenkel WpEn:Abraham_Fraenkel }

 ZFC집합론 ZFC_set_theory
  ZF set theory에 [[선택공리,choice_axiom]]{ https://encyclopediaofmath.org/wiki/Axiom_of_choice }만 추가한?
  https://encyclopediaofmath.org/wiki/ZFC
  https://ncatlab.org/nlab/show/ZFC

 NBG집합론 NBG_set_theory
  von Neumann–Bernays–Gödel set theory (NBG)
  ZF의 [[확장,extension]], esp [[conservative_extension]] { conservative extension WpKo:보존적_확장 WpEn:Conservative_extension Up: [[보존,conservation]] [[확장,extension]] }
  [[https://terms.naver.com/entry.naver?docId=4125138&cid=60207&categoryId=60207 수학백과 NBG 집합론]]
  [[WpEn:von_Neumann–Bernays–Gödel_set_theory]]
  [[WpJa:フォン・ノイマン＝ベルナイス＝ゲーデル集合論]]
  ... Ndict:nbg+set+theory Ggl:nbg+set+theory

 set_theoretic_programming
 '''set theoretic programming'''
 집합론적 프로그래밍
 [[WpEn:Set_theoretic_programming]]
 Ggl:"set theoretic programming"
 Up: [[집합론,set_theory]] [[프로그래밍패러다임,programming_paradigm]]

Related/MKL:
[[논리,logic]] esp [[수리논리,mathematical_logic]]
[[사건,event]]
[[수학기초론]]
 foundations of mathematics
 WpKo:수학기초론
 WpEn:Foundations_of_mathematics 
 WpJa:数学基礎論
[[공리,axiom]]
[[axiom_system]] 공리계 maybe... or 공리체계
[[universe]] =,universe { 전체 .... KmsE:universe }
[[모형,model]] [[모델,model]]


Topics:
[[reflection_principle]] - writing

<<tableofcontents>>

= 연산 =
union
 $A\cup B=\left{x\middle|x\in A\textrm{ or }x\in B\right}$
intersection
 $A\cap B=\left{x\middle|x\in A\textrm{ and }x\in B\right}$
 mutually exclusive or disjoint:
  $A\cap B=\not\bigcirc$
[[부분집합,subset]]
complement
 $A^C=\left{x\middle|x\not\in A\right}$

[[교환법칙,commutativity]]
 $A\cap B=B\cap A$
 $A\cup B=B\cup A$
[[결합법칙,associativity]]
 $A\cup(B\cup C)=(A\cup B)\cup C$
 $A\cap(B\cap C)=(A\cap B)\cap C$
[[분배법칙,distributivity]]
 $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$
 $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$
[[드모르간_법칙,De_Morgan_law]]
{
'''De Morgan's laws'''
$(A\cup B)^C=A^C\cap B^C$
$(A\cap B)^C=A^C\cup B^C$

$\neg(p\wedge q)\Leftrightarrow \neg p \vee \neg q$
$\neg(p\vee q)\Leftrightarrow \neg p \wedge \neg q$
이것은 [[논리곱]] ∧과 [[논리합]] ∨사이의 [[쌍대성,duality]].[* [[https://terms.naver.com/entry.naver?docId=3338026&cid=47324&categoryId=47324 수학백과: 부정명제]]의 3.1.]

from (10개의 특강으로 끝내는 수학의 기본 원리)
{
||$\sim(A\cup B)=(\sim A)\cap(\sim B)$ ||합집합의 여집합은 여집합들의 교집합이다. ||
||$\sim(A\cap B)=(\sim A)\cup(\sim B)$ ||교집합의 여집합은 여집합들의 합집합이다. ||
}

[[WpKo:드_모르간의_법칙]]
[[WpEn:De_Morgan's_laws]]
... Ndict:"드 모르간 법칙" Ggl:"드 모르간 법칙"
Up: [[명제논리,propositional_logic]], [[집합론,set_theory]] [[법칙,law]]
}

= Bmks ko =
계승혁 - 집합과 수리논리 강의 동영상 (2020)
https://www.math.snu.ac.kr/~kye/lecture_V/V_set/index.html

= Bmks en =
Set theory symbols
https://mathvault.ca/hub/higher-math/math-symbols/set-theory-symbols/

[[WpEn:List_of_set_identities_and_relations]] = https://en.wikipedia.org/wiki/List_of_set_identities_and_relations

집합론의 역사 관련 (tmp)
The Early Development of Set Theory (Stanford Encyclopedia of Philosophy)
https://plato.stanford.edu/entries/settheory-early/

----
Twins:
 [[wiki:Google:집합론]]
 https://en.citizendium.org/wiki/Set_theory
 https://planetmath.org/settheory
 https://mathworld.wolfram.com/SetTheory.html
 https://proofwiki.org/wiki/Definition:Set_Theory
https://ncatlab.org/nlab/show/set+theory
https://encyclopediaofmath.org/wiki/Set_theory
https://everything2.com/title/set+theory
https://plato.stanford.edu/entries/set-theory/
[[Wiki:SetTheory]]
[[https://terms.naver.com/entry.naver?docId=3405344&cid=47324&categoryId=47324 수학백과 집합론]]