Sets: Hierarchy

• 1. Set concepts

  2. Elementary collections

  3. unstructured collections

     • set

     • bag, multiset

     • list, sequence

     • (unordered list)

  4. notion of sets

  5. fundamental set properties

     • disorder

     • uniqueness of elements

       • multiplicity of elements

  6. set (member) element vs object

  7. Set representation

     • intensional representation

     • extensional representation

  8. Set notation

     • Roster form

     • Set builder notation

• Set cardinality

  • Finite set

    • the empty set vs nonempty set

      • the empty set

      • nonempty set

    • singleton set

    • unordered pair

  • Infinite sets

    • Infinite enumerable set

    • Infinite innumerable set

  • Set equality

    • intensional equality

    • extensional equality

    • equal sets

    • equivalent sets

    • equinumerousity

  • Universal set

• Operations on sets

  • Set union

  • Set intersection

  • Set difference

  • Set complement (negation)

  • Cartesian product, cross/dot product

• Set properties and operations

  • Cardinal number

  • Unordered pair

  • Ordered pair

  • Set covering

  • Set partitioning

  • Number of proper subsets

  • Bell Numbers

  • Natural numbers in terms of sets

• Algebraic properties of sets

  • Set identities

  • Idempotency

• Set relations

  • inclusion relation

    • subset, superset

    • proper subset, proper superset

  • membership relation

  • indexed set

  • family of sets

• Basic set relations

  • Set membership

    • Set member

    • Set paradox

    • Richard's paradox

    • Russell's paradox

  • Set inclusion

    • Disjoint sets

    • pair-wise disjoint sets

    • Overlapping sets

    • subset, superset

    • proper subset, proper superset

  • Power set

  • set product, Cartesian product

  • ordered pair

• Fundamental sets

  • The set of natural numbers

  • The set of integers

  • The set of rational numbers

  • The set of real numbers

  • The set of complex numbers

• Mathematical theories

  • Set theory

    • naive set theory

    • axiomatic set theory

    • axioms of set theory

  • Relation theory

  • Order theory

  • Domain theory

  • Function theory