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Set Theory Basics

36 cards · 🇺🇸 English

Sets, subsets, unions, intersections and set operations.

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Set theory

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  1. 1Set theory
    The branch of mathematical logic that studies sets, which can be informally described as collections of objects
  2. 2What notation is used to indicate membership in a set?
    o ∈ A means that object o is a member (or element) of set A
  3. 3How is a set described?
    By listing elements separated by commas within braces { }, or by a characterizing property of its elements
  4. 4Subset relation
    If all members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B
  5. 5Proper subset
    A is a proper subset of B if A is a subset of B but A is not equal to B
  6. 6Is a set a subset of itself?
    Yes, every set is a subset of itself (though not a proper subset)
  7. 7Union of two sets
    A ∪ B is the set of all objects that are a member of A, or B, or both
  8. 8Intersection of two sets
    A ∩ B is the set of all objects that are members of both A and B
  9. 9Union of {1, 2, 3} and {2, 3, 4}
    {1, 2, 3, 4}
  10. 10Intersection of {1, 2, 3} and {2, 3, 4}
    {2, 3}
  11. 11Set difference
    U ∖ A is the set of all members of U that are not members of A
  12. 12Set difference {1, 2, 3} ∖ {2, 3, 4}
    {1}
  13. 13Complement of a set A in universe U
    When A is a subset of U, the complement is U ∖ A, sometimes denoted Ac
  14. 14Symmetric difference
    A △ B is the set of all objects that are a member of exactly one of A and B (in one set, but not both)
  15. 15Symmetric difference of {1, 2, 3} and {2, 3, 4}
    {1, 4}
  16. 16Cartesian product
    A × B is the set whose members are all possible ordered pairs (a, b), where a is in A and b is in B
  17. 17Cartesian product of {1, 2} and {red, white}
    {(1, red), (1, white), (2, red), (2, white)}
  18. 18Empty set
    The unique set containing no elements, denoted { }, ∅, or ⌀
  19. 19Power set
    The power set of A, denoted P(A), is the set whose members are all possible subsets of A
  20. 20Power set of {1, 2}
    { { }, {1}, {2}, {1, 2} }
  21. 21Cantor's revolutionary discovery
    That the set of all real numbers is uncountable—one cannot put all real numbers in a list
  22. 22Cantor's theorem
    The size of the power set of A is strictly larger than the size of A, even when A is infinite
  23. 23What notation did Cantor use for cardinal numbers?
    The Hebrew letter ℵ (aleph) with a natural number subscript
  24. 24What notation did Cantor use for ordinal numbers?
    The Greek letter ω (omega)
  25. 25Russell's paradox
    Let R be the set of all sets not members of themselves. If R is not in itself, its definition says it is; if R is in itself, then it's not (since R contains only sets not in themselves)
  26. 26Naive set theory
    The non-formalized systems of set theory investigated in the early stage before paradoxes were discovered
  27. 27Most widely studied axiomatic set theory
    Zermelo–Fraenkel set theory with the axiom of choice (ZFC)
  28. 28Pure set
    A set where all of its members are sets, all members of its members are sets, and so on
  29. 29Von Neumann universe
    The universe of pure sets organized into a cumulative hierarchy based on how deeply their members are nested
  30. 30What rank is assigned to the empty set in the von Neumann hierarchy?
    Rank 0

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