{
    "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."
        }
    ]
}