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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 />