Set operations<br />with special thanks to Georg Cantor<br />
Thank you for set theory<br />Georg Cantor (1845 – 1918)<br />
Founded set theory<br />A single paper in 1874<br />
Basic operations<br />Binary relations and more<br />
Fundamental relation<br />An object o <br />A set A<br />
relation<br />If o is an element of A, we write o ∈ A.<br />
inclusion<br />Let A be a set of all chihuahuas<br />
inclusion<br />Let B be the setof all dogs<br />
inclusion<br />All Chihuahuas are dogs. <br />
inclusion<br />A is a subset of B. <br />
inclusion<br />B is a superset of A.<br />
Inclusion<br />Subsets and supersets<br /> If all the members of set A are also members of set B, then A is a subset of B,...
union<br />Union of two sets: <br />A ∪ B<br />
union<br />denoted A ∪ B<br />
Union<br />of the sets A and B is the set of all objects that are a member of A, or B, or both.<br />
Union<br />The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .<br />
Intersection<br />
intersection<br />Intersection of three sets: <br />A ∩ B ∩ C<br />
intersection<br />denoted A ∩ B<br />
Intersection<br />of the sets A and B is the set of all objects that are members of both A and B. <br />
Intersection<br />The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .<br />
complement<br />Ac = UA<br />
complement<br />The relative complement of A (left circle) inB (right circle):<br />
complement<br />denoted Ac<br />
complement<br />is the set of all members of U that are not members of A.<br />
complement<br />The complement of {1,2,3} relative to {2,3,4} is {4}<br />
Cartesian product<br />Direct product of two sets <br />
Cartesian product<br />denoted A × B<br />
Cartesian product<br />
Cartesian product<br />of X and Y is the set whose members are all possible ordered pairs (x , y) where x is a member of X...
Power set<br />The set of all subsets of a set<br />
Power set<br />P(S) or ℘(S)<br />
Power set<br />If S is the set {x, y, z} <br />	the power set is <br />	P(S)={∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x...
Power set<br />the powerset of {1, 2} is { {}, {1}, {2}, {1,2} }<br />
Attribution<br />http://en.wikipedia.org/<br />
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On set operations

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On set operations

  1. 1. Set operations<br />with special thanks to Georg Cantor<br />
  2. 2. Thank you for set theory<br />Georg Cantor (1845 – 1918)<br />
  3. 3. Founded set theory<br />A single paper in 1874<br />
  4. 4. Basic operations<br />Binary relations and more<br />
  5. 5. Fundamental relation<br />An object o <br />A set A<br />
  6. 6. relation<br />If o is an element of A, we write o ∈ A.<br />
  7. 7. inclusion<br />Let A be a set of all chihuahuas<br />
  8. 8. inclusion<br />Let B be the setof all dogs<br />
  9. 9. inclusion<br />All Chihuahuas are dogs. <br />
  10. 10. inclusion<br />A is a subset of B. <br />
  11. 11. inclusion<br />B is a superset of A.<br />
  12. 12. Inclusion<br />Subsets and supersets<br /> If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B.<br />
  13. 13. union<br />Union of two sets: <br />A ∪ B<br />
  14. 14. union<br />denoted A ∪ B<br />
  15. 15. Union<br />of the sets A and B is the set of all objects that are a member of A, or B, or both.<br />
  16. 16. Union<br />The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .<br />
  17. 17. Intersection<br />
  18. 18. intersection<br />Intersection of three sets: <br />A ∩ B ∩ C<br />
  19. 19. intersection<br />denoted A ∩ B<br />
  20. 20. Intersection<br />of the sets A and B is the set of all objects that are members of both A and B. <br />
  21. 21. Intersection<br />The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .<br />
  22. 22. complement<br />Ac = UA<br />
  23. 23. complement<br />The relative complement of A (left circle) inB (right circle):<br />
  24. 24. complement<br />denoted Ac<br />
  25. 25. complement<br />is the set of all members of U that are not members of A.<br />
  26. 26. complement<br />The complement of {1,2,3} relative to {2,3,4} is {4}<br />
  27. 27. Cartesian product<br />Direct product of two sets <br />
  28. 28. Cartesian product<br />denoted A × B<br />
  29. 29. Cartesian product<br />
  30. 30. Cartesian product<br />of X and Y is the set whose members are all possible ordered pairs (x , y) where x is a member of X and y is a member of Y<br />
  31. 31. Power set<br />The set of all subsets of a set<br />
  32. 32. Power set<br />P(S) or ℘(S)<br />
  33. 33. Power set<br />If S is the set {x, y, z} <br /> the power set is <br /> P(S)={∅, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}<br />
  34. 34. Power set<br />the powerset of {1, 2} is { {}, {1}, {2}, {1,2} }<br />
  35. 35. Attribution<br />http://en.wikipedia.org/<br />

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