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Worksheet on Sets - Solutions

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Much of mathematics, including probability and statistics, is ultimately based upon set theory. So it is important to have a thorough understanding of this topic. Set theory is the study of collections of objects called elements. These elements can be combined using set operations of union, intersection and complement.

Given the sets A = {1, 2, 3, 4, 5}, B = {6, 7, 8, 9, 10}, C = {1, 3, 5, 7, 9}, D = {2, 4, 6, 8, 10}, E = {3, 4, 5, 6, 7}, F = {3, 4, 5},and universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, test your knowledge about sets by stating which of the following are true and which are false. Here is a version of the worksheet, but without solutions.

  1. A = B
  2. SOLUTION: False. None of the elements in the set A are elements of the set B. These sets are not equal.
  3. A = A
  4. SOLUTION: True. Every set is equal to itself.
  5. CE
  6. SOLUTION: False. C is not a subset of E because 1 is an element of C that is not an element of E.
  7. FE
  8. SOLUTION: True. All of the elements of F are also elements of the set E. Therefore F is a subset of E.
  9. EF
  10. SOLUTION: False. E is not a subset of F because 6 is an element of E that is not an element of F.
  11. DU
  12. SOLUTION: True. Any set is a subset of the universal set.
  13. Ø ⊆ F
  14. SOLUTION: True. The empty set is a subset of any set.
  15. Ø ⊆ Ø
  16. SOLUTION: True. Since the empty set is a subset of any set, it follows that it is also a subset of itself. In general every set is a subset of itself.
  17. Ø ∈ A
  18. SOLUTION: False. Although the empty set is a subset of A, the empty set is not an element of the set A. The elements of A are whole numbers from one to five, and the empty set is not any of these elements.
  19. 1 ∈ A
  20. SOLUTION: True. The number one is an element of the set A.
  21. 7 ∈ A
  22. SOLUTION: False. The number 7 is not an element of the set A.
  23. 3 ∉ D
  24. SOLUTION: True. The number 3 is not an element of the set D.
  25. 3 ∉ A
  26. SOLUTION: False. The number three is an element of the set A. Therefore the statement "3 is not an element of the set A" is false.
  27. F ∈ E
  28. SOLUTION: False. The elements of the set E consist of three, four, five, six and seven. Even though F is a collection of some of these, the set F is not an element of E. The set F is a subset of the set E.
  29. 1 ∉ B
  30. SOLUTION: True. One is not an element of the set B.
  31. Ø ∈ Ø
  32. SOLUTION: False. This one is a bit tricky. The empty set is a subset of any set, but it is not an element of itself.
  33. Ø ∈ {Ø}
  34. SOLUTION: True. This one is trickier yet. The set on the right has the empty set as an element. Therefore this is a true statement.
  35. AC = B
  36. SOLUTION: True. The complement of the set A is the set B because B is the set of all elements in the universal set U that are not elements of the set A.
  37. ØC = U
  38. SOLUTION: True. This involves a careful thinking through of the definition of empty set and complement. The empty set is the set with no elements. The complement of the empty set is everything in the universal set U that is not an element of the empty set, so therefore the complement of the empty set is the universal set.
  39. AB = C
  40. SOLUTION: False. The set of elements in the set A or the set B is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. This is not equal to the set C.
  41. AB = C
  42. SOLUTION: False. The sets A and B share no elements in common. Therefore their intersection is the empty set.
  43. AB = U
  44. SOLUTION: True. The set of elements in either A or B is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and this is the universal set U.
  45. AE = F
  46. SOLUTION: True. The numbers three, four and five are the only elements that are common to both sets A and E. Since these are the same elements in the set F, the above equation is true.
  47. BC = {7, 9}
  48. SOLUTION: False. The union of two sets B and C is the set of all elements in at least one of these two sets, and possibly both. The correct union BC = {1, 3, 5, 6, 7, 8, 9, 10}.
  49. BC = {7, 9}
  50. SOLUTION: True. The only elements that B and C share in common are the numbers seven and nine.
  51. BF = Ø
  52. SOLUTION: True. The sets B and F share no elements in common. Therefore their intersection is the empty set.
  53. F ∪ Ø = F
  54. SOLUTION: True. The union of any set with the empty set is the original set. This is because there are no elements from the empty set being added to the set F.
  55. E ∩ Ø = Ø
  56. SOLUTION: True. The intersection of any set with the empty set will result in the empty set.
  57. DU = D
  58. SOLUTION: False. The union of the universal set with any set will result in the universal set U.
  59. (AB )C = A CB C
  60. SOLUTION: True. This is an instance of a set theory equation known as DeMorgan’s Law.

So how did you do? If you got less than 20 of them right, try again.

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