api: YAML JSON TOON Database
version: 1.0.0
format: json
dataset:
  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:
    - You need the notation and meaning of a set operation.
    - You are translating between set language and logic or SQL.
  caveats:
    - 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:
    - combinatorics-fundamentals
    - statistics-basics
    - sql-join-types
  category: Mathematics
  category_slug: mathematics
  tags: math,set-theory,logic
  view_count: 0
  entry_count: 14
  fields:
    - section: data
      field: concept
      type: string
      example: Union
    - section: data
      field: notation
      type: string
      example: A ∪ B
    - section: data
      field: description
      type: string
      example: Elements in A or B or both.
data:
  - concept: Union
    notation: A ∪ B
    description: Elements in A or B or both.
  - concept: Intersection
    notation: A ∩ B
    description: Elements common to A and B.
  - concept: Complement
    notation: A'
    description: Elements not in A within the universal set.
  - concept: Subset
    notation: A ⊆ B
    description: Every element of A is also in B.
  - concept: Superset
    notation: A ⊇ B
    description: A contains all elements of B.
  - concept: Power Set
    notation: P(A)
    description: Set of all subsets of A.
  - concept: Cartesian Product
    notation: A × B
    description: Set of all ordered pairs from A and B.
  - concept: Set Difference
    notation: A ∖ B
    description: Elements in A that are not in B.
  - concept: Symmetric Difference
    notation: A Δ B
    description: Elements in exactly one of A or B.
  - concept: Empty Set
    notation: ∅
    description: The set with no elements; it is a subset of every set.
  - concept: Cardinality
    notation: "|A|"
    description: The number of elements in A.
  - concept: Element Of
    notation: x ∈ A
    description: x is a member of A.
  - concept: Disjoint Sets
    notation: A ∩ B = ∅
    description: Sets with no elements in common.
  - concept: De Morgan's Laws
    notation: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
    description: The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.