Sets and Set Operations
Set – A collection of objects     example:      a set of tiresElement – An object contained within a set    example:    my...
Finite set – Contains a                        countable number of                        objects                        E...
Cardinal Number –                          Used to count the                          objects in a set                    ...
Equal sets – Sets that contain exactly the sameelements (in any order)  {A, R, T, S} = {S, T, A, R}Notation: A = B means s...
Equivalent sets – Sets that              contain the same number              of elements (elements do              not ha...
Empty Set – A set that contains no elements Notation: { } or                  Universal Set – A set that                  ...
Complement of a set – A set that contains all ofthe elements of the universal set that are not ina given setNotation:   B ...
A = {2, 4, 6, 8}                B = {1, 2, 3, 4, …}C = {1, 2, 3, 4, 5}             D={}E = {Al, Ben, Carl, Doug}       F =...
U = {1, 2, 3, 4, 5, 6, 7}M = {2, 4, 6}What is M ? {1, 3, 5, 7}
Is { } the same as     ? YesIs {   } the same as     ? No
Set B is a subset of set A if every element ofset B is also an element of set A.     Notation: B AW = {1, 2, 3, 4, 5}     ...
Set B is a proper subset of set A if everyelement of set B is also an element of set AAND B is not equal to A.     Notatio...
How many subsets can a set have?               Number of                          Number of   Set         Elements        ...
How many proper subsets can a set have?                                        Number of               Number of          ...
W = {a, b, c, d, e, f}How many subsets does set W have?    26 = 64How many proper subsets does set W have?    26 – 1 = 64 ...
A Venn Diagram allows us to organize theelements of a set according to theirattributes.
U = {1, 2, 3, 4, 5, 6.5}even                   1    odd        4                    3            2        5 6.5           ...
small                               blue                                     triangleNational Library of Virtual Manipulat...
Set OperationsThe intersection of sets A and B is the set of allelements in both sets A and B               notation: A B
The union of sets A and B is the set of allelements in either one or both of sets A and B               notation: A B
The union of sets A and B is the set of allelements in either one or both of sets A and B               notation: A B
The union of sets A and B is the set of allelements in either one or both of sets A and B               notation: A B
A = {1, 2, 3, 4, 5}   B = {2, 4, 6}   C = {3, 5, 7}A    B = {2, 4}                  A – B = {1, 3, 5}A    B = {1, 2, 3, 4,...
Representing sets with Venn diagrams      A                 B              A                       B                      ...
A       B   A       B    A           A
A         B     A               B    AUB                 A   B      A             B              A B
A         B   A           B    AUB           A   B
A             B          C(A U B)       C
A=   {1, 2, 4, 5}            A                       BB=   {2, 3, 5, 6}                1       2       3C=   {4, 5, 6, 7} ...
A = {3, 6, 7, 8}                 A                       BB = {2, 3, 5, 6}                     1       2       3C = {4, 5,...
A                        B                1              3                         2                                      ...
20                      Out of 20 students: B                F       8 play baseball                          7 play footb...
30                    Out of 30 people surveyed:                             20 like Blue  B                P         20 l...
1150 day 2
1150 day 2
1150 day 2
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1150 day 2

  1. 1. Sets and Set Operations
  2. 2. Set – A collection of objects example: a set of tiresElement – An object contained within a set example: my car’s left front tire
  3. 3. Finite set – Contains a countable number of objects Example: The car has 4 tiresInfinite set- Contains an unlimited number ofobjects Example: The counting numbers {1, 2, 3, …}
  4. 4. Cardinal Number – Used to count the objects in a set Example: There are 26 letters in the alphabetOrdinal Number – Used to describe the positionof an element in a set Example: The letter D is the 4th letter of the alphabet
  5. 5. Equal sets – Sets that contain exactly the sameelements (in any order) {A, R, T, S} = {S, T, A, R}Notation: A = B means set A equals set B
  6. 6. Equivalent sets – Sets that contain the same number of elements (elements do not have to be the same) {C, A, T} ~ {d, o, g}Notation: A ~ B means set A is equivalent to set B
  7. 7. Empty Set – A set that contains no elements Notation: { } or Universal Set – A set that contains all of the elements being considered Notation: U
  8. 8. Complement of a set – A set that contains all ofthe elements of the universal set that are not ina given setNotation: B means the complement of B
  9. 9. A = {2, 4, 6, 8} B = {1, 2, 3, 4, …}C = {1, 2, 3, 4, 5} D={}E = {Al, Ben, Carl, Doug} F = { 5, 4, 3, 2, 1}G = {x | x < 6 and x is a counting number} Set Builder Notation {1, 2, 3, 4, 5}Which sets are finite? n(E) = 4 A, C, D, E, F, G n(G) = 5Which sets are equal to set C? F, GWhich sets are equivalent to set A? E
  10. 10. U = {1, 2, 3, 4, 5, 6, 7}M = {2, 4, 6}What is M ? {1, 3, 5, 7}
  11. 11. Is { } the same as ? YesIs { } the same as ? No
  12. 12. Set B is a subset of set A if every element ofset B is also an element of set A. Notation: B AW = {1, 2, 3, 4, 5} X = {1, 3, 5}Y = {2, 4, 6} Z = {4, 2, 1, 5, 3}True or False: X W True Y W False Z W True The empty set is a W True subset of every set
  13. 13. Set B is a proper subset of set A if everyelement of set B is also an element of set AAND B is not equal to A. Notation: B AW = {1, 2, 3, 4, 5} X = {1, 3, 5}Y = {2, 4, 6} Z = {4, 2, 1, 5, 3}True or False: X W True Y W False Z W False
  14. 14. How many subsets can a set have? Number of Number of Set Elements Subsets Subsets {a} 1 {a},{ } 2 {a, b} 2 {a},{b},{a,b},{ } 4 {a, b, c} 3 {a},{b},{c},{a,b}, {a,c},{b,c},{a,b,c}, 8 { }{a, b, c, d} 4 16 n 2n If a set has n elements, it has 2n subsets
  15. 15. How many proper subsets can a set have? Number of Number of Proper Set Elements Proper Subsets Subsets {a} 1 X {a},{ } 1 {a, b} 2 X {a},{b},{a,b},{ } 3 {a, b, c} 3 {a},{b},{c},{a,b}, X {a,c},{b,c},{a,b,c}, 7 { }{a, b, c, d} 4 15 n 2n – 1 If a set has n elements, it has 2n – 1 proper subsets
  16. 16. W = {a, b, c, d, e, f}How many subsets does set W have? 26 = 64How many proper subsets does set W have? 26 – 1 = 64 – 1 = 63
  17. 17. A Venn Diagram allows us to organize theelements of a set according to theirattributes.
  18. 18. U = {1, 2, 3, 4, 5, 6.5}even 1 odd 4 3 2 5 6.5 prime
  19. 19. small blue triangleNational Library of Virtual Manipulatives Attribute Blocks
  20. 20. Set OperationsThe intersection of sets A and B is the set of allelements in both sets A and B notation: A B
  21. 21. The union of sets A and B is the set of allelements in either one or both of sets A and B notation: A B
  22. 22. The union of sets A and B is the set of allelements in either one or both of sets A and B notation: A B
  23. 23. The union of sets A and B is the set of allelements in either one or both of sets A and B notation: A B
  24. 24. A = {1, 2, 3, 4, 5} B = {2, 4, 6} C = {3, 5, 7}A B = {2, 4} A – B = {1, 3, 5}A B = {1, 2, 3, 4, 5, 6} C – A = {7}C B = {2, 3, 4, 5, 6, 7)C B= { }The set complement X – Y is the set of allelements of X that are not in Y
  25. 25. Representing sets with Venn diagrams A B A B 1 2 3 1 2 3 5 4 6 4 7 8 Two attributes C 22 or 4 regions Three attributes 23 or 8 regions
  26. 26. A B A B A A
  27. 27. A B A B AUB A B A B A B
  28. 28. A B A B AUB A B
  29. 29. A B C(A U B) C
  30. 30. A= {1, 2, 4, 5} A BB= {2, 3, 5, 6} 1 2 3C= {4, 5, 6, 7} 5 4 6C= {1, 2, 3, 8} 7 8A U B = {1, 2, 3, 4, 5, 6} C(A U B) C = {1, 2, 3} (A U B) C
  31. 31. A = {3, 6, 7, 8} A BB = {2, 3, 5, 6} 1 2 3C = {4, 5, 6, 7} 4 5 6 7B C = {5, 6} 8 CA U (B C) = {3, 5, 6, 7, 8} A U (B C)
  32. 32. A B 1 3 2 4How many stars are in: Circle A 3 Either A or B 7 Circle B 5 Exactly one circle 6 Only Circle A 2 Neither circle 2 Both A and B 1 Total stars = 9
  33. 33. 20 Out of 20 students: B F 8 play baseball 7 play football 5 3 4 3 play both sports 8How many play neither sport? 8How many play only baseball? 5How many play exactly one sport? 5 + 4 = 9
  34. 34. 30 Out of 30 people surveyed: 20 like Blue B P 20 like Pink 4 4 5 15 like Green 10 14 like Blue and Pink 2 1 11 like Pink and Green 2 12 like Blue and Green 2 G 10 like all 3 colorsHow many people like only Pink? 5How many like Blue and Green but not Pink? 2How many like none of the 3 colors? 2How many like exactly two of the colors? 4 + 2 + 1 =7
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