api: YAML JSON TOON Database
version: 1.0.0
format: toon
dataset:
  id: 2891
  slug: set-theory-basics
  title: Set Theory Basics
  description: Fundamental set operations and relations.
  summary: "The core vocabulary and notation of set theory: union, intersection, complement, subsets, power sets and Cartesian products, with set difference, symmetric difference, cardinality, membership, disjoint sets and De Morgan's laws."
  use_when[2]: You need the notation and meaning of a set operation.,You are translating between set language and logic or SQL.
  caveats[2]: Complement is relative to a stated universal set; without one it is undefined.,"Notations differ between textbooks (A^c, A', and a bar over A all denote complement)."
  related[3]: combinatorics-fundamentals,statistics-basics,sql-join-types
  entry_count: 14
  fields[3]{section,field,type,example}:
    data,concept,string,Union
    data,notation,string,A ∪ B
    data,description,string,Elements in A or B or both.
  category: Mathematics
  category_slug: mathematics
  tags: "math,set-theory,logic"
  view_count: 71
  created_at: 1793403093
  updated_at: 1793403093
data[14]{concept,notation,description}:
  Union,A ∪ B,Elements in A or B or both.
  Intersection,A ∩ B,Elements common to A and B.
  Complement,A',Elements not in A within the universal set.
  Subset,A ⊆ B,Every element of A is also in B.
  Superset,A ⊇ B,A contains all elements of B.
  Power Set,P(A),Set of all subsets of A.
  Cartesian Product,A × B,Set of all ordered pairs from A and B.
  Set Difference,A ∖ B,Elements in A that are not in B.
  Symmetric Difference,A Δ B,Elements in exactly one of A or B.
  Empty Set,∅,The set with no elements; it is a subset of every set.
  Cardinality,|A|,The number of elements in A.
  Element Of,x ∈ A,x is a member of A.
  Disjoint Sets,A ∩ B = ∅,Sets with no elements in common.
  De Morgan's Laws,(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ,"The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements."