Math 1300: Section 7-2 Sets

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Math 1300: Section 7-2 Sets

  1. 1. Set Properties and Set Notation Set OperationsMath 1300 Finite Mathematics Section 7-2: Sets Department of Mathematics University of Missouri October 16, 2009 university-logo Math 1300 Finite Mathematics
  2. 2. Set Properties and Set Notation Set OperationsA set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection. university-logo Math 1300 Finite Mathematics
  3. 3. Set Properties and Set Notation Set OperationsA set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets. university-logo Math 1300 Finite Mathematics
  4. 4. Set Properties and Set Notation Set OperationsA set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets.Each object in a set is called a member or element of theset. university-logo Math 1300 Finite Mathematics
  5. 5. Set Properties and Set Notation Set OperationsSymbolically, a∈A means "a is an element of set A" a∈A means "a is not an element of set A" university-logo Math 1300 Finite Mathematics
  6. 6. Set Properties and Set Notation Set OperationsSymbolically, a∈A means "a is an element of set A" a∈A means "a is not an element of set A"A set without any elements is called the empty, or null,set. university-logo Math 1300 Finite Mathematics
  7. 7. Set Properties and Set Notation Set OperationsSymbolically, a∈A means "a is an element of set A" a∈A means "a is not an element of set A"A set without any elements is called the empty, or null,set.Symbollically ∅ denotes the empty set university-logo Math 1300 Finite Mathematics
  8. 8. Set Properties and Set Notation Set OperationsA set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set. university-logo Math 1300 Finite Mathematics
  9. 9. Set Properties and Set Notation Set OperationsA set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.Thus, if P(x) is a statement about x, thenS = {x|P(x)} means “S is the set of all x such that P(x) is true” university-logo Math 1300 Finite Mathematics
  10. 10. Set Properties and Set Notation Set OperationsA set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.Thus, if P(x) is a statement about x, thenS = {x|P(x)} means “S is the set of all x such that P(x) is true”Example: Rule Listing {x|x is a weekend day} = {Saturday, Sunday} {x|x 2 = 4} = {−2, 2} {x|x is a positive odd number} = {1, 3, 5, . . . } university-logo Math 1300 Finite Mathematics
  11. 11. Set Properties and Set Notation Set OperationsA set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.Thus, if P(x) is a statement about x, thenS = {x|P(x)} means “S is the set of all x such that P(x) is true”Example: Rule Listing {x|x is a weekend day} = {Saturday, Sunday} {x|x 2 = 4} = {−2, 2} {x|x is a positive odd number} = {1, 3, 5, . . . }The first two sets in this example are finite sets; the last set isan infinite set. university-logo Math 1300 Finite Mathematics
  12. 12. Set Properties and Set Notation Set OperationsWhen listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter. university-logo Math 1300 Finite Mathematics
  13. 13. Set Properties and Set Notation Set OperationsWhen listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B. university-logo Math 1300 Finite Mathematics
  14. 14. Set Properties and Set Notation Set OperationsWhen listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal. university-logo Math 1300 Finite Mathematics
  15. 15. Set Properties and Set Notation Set OperationsWhen listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically, A ⊂ B means “A is a subset of B” A = B means “A equals B” A ⊂ B means “A is not a subset of B” A = B means “A and B do not have the same elements” university-logo Math 1300 Finite Mathematics
  16. 16. Set Properties and Set Notation Set OperationsWhen listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically, A ⊂ B means “A is a subset of B” A = B means “A equals B” A ⊂ B means “A is not a subset of B” A = B means “A and B do not have the same elements”∅ is a subset of every set. university-logo Math 1300 Finite Mathematics
  17. 17. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}. university-logo Math 1300 Finite Mathematics
  18. 18. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A. university-logo Math 1300 Finite Mathematics
  19. 19. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A. university-logo Math 1300 Finite Mathematics
  20. 20. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A.What one-element subsets? university-logo Math 1300 Finite Mathematics
  21. 21. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A.What one-element subsets? {b}, {c} and {d}. university-logo Math 1300 Finite Mathematics
  22. 22. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A.What one-element subsets? {b}, {c} and {d}.What two-element subsets are there? university-logo Math 1300 Finite Mathematics
  23. 23. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A.What one-element subsets? {b}, {c} and {d}.What two-element subsets are there? {b, c}, {b, d}, {c, d} university-logo Math 1300 Finite Mathematics
  24. 24. Set Properties and Set Notation Set OperationsExample: List all subsets of the set A = {b, c, d}.First, ∅ ⊂ A.Also, every set is a subset of itself, so A ⊂ A.What one-element subsets? {b}, {c} and {d}.What two-element subsets are there? {b, c}, {b, d}, {c, d}There are a total of 8 subsets of set A. university-logo Math 1300 Finite Mathematics
  25. 25. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: university-logo Math 1300 Finite Mathematics
  26. 26. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B university-logo Math 1300 Finite Mathematics
  27. 27. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C university-logo Math 1300 Finite Mathematics
  28. 28. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B university-logo Math 1300 Finite Mathematics
  29. 29. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A university-logo Math 1300 Finite Mathematics
  30. 30. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A C⊂A university-logo Math 1300 Finite Mathematics
  31. 31. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A C⊂A B⊂A university-logo Math 1300 Finite Mathematics
  32. 32. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A C⊂A B⊂A ∅⊂A university-logo Math 1300 Finite Mathematics
  33. 33. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A C⊂A B⊂A ∅⊂A ∅⊂C university-logo Math 1300 Finite Mathematics
  34. 34. Set Properties and Set Notation Set OperationsExample: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} andC = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements istrue: A=B A⊂C A⊂B C=A C⊂A B⊂A ∅⊂A ∅⊂C ∅∈A university-logo Math 1300 Finite Mathematics
  35. 35. Set Properties and Set Notation Set OperationsDefinitionThe union of two sets A and B is the set of all elements formedby combining all the elements of set A and all the elements ofset B into one set. It is written A ∪ B. Symbolically, A ∪ B = {x|x ∈ A or x ∈ B} U A B university-logo Math 1300 Finite Mathematics
  36. 36. Set Properties and Set Notation Set OperationsDefinitionThe intersection of two sets A and B is the set of all elementsthat are common to both A and B. It is written A ∩ B.Symbolically, A ∩ B = {x|x ∈ A and x ∈ B} U A B university-logo Math 1300 Finite Mathematics
  37. 37. Set Properties and Set Notation Set OperationsIf two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅. university-logo Math 1300 Finite Mathematics
  38. 38. Set Properties and Set Notation Set OperationsIf two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅.The set of all elements under consideration is called theuniversal set U. Once the universal set is determined for aparticular problem, all other sets under consideration mustbe subsets of U. university-logo Math 1300 Finite Mathematics
  39. 39. Set Properties and Set Notation Set OperationsDefinitionThe complement of a set A is defined as the set of elementsthat are contained in U, the universal set, but not contained inset A. We donte the complement by A . Symbolically, A = {x ∈ U|x ∈ A} U A B university-logo Math 1300 Finite Mathematics
  40. 40. Set Properties and Set Notation Set OperationsExample: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M . university-logo Math 1300 Finite Mathematics
  41. 41. Set Properties and Set Notation Set OperationsExample: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M .The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩ M ). university-logo Math 1300 Finite Mathematics
  42. 42. Set Properties and Set Notation Set OperationsExample: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M .The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩ M ).The study is based on 1000 commuters, so n(U) = 1000. university-logo Math 1300 Finite Mathematics
  43. 43. Set Properties and Set Notation Set OperationsExample: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M .The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩ M ).The study is based on 1000 commuters, so n(U) = 1000.The number of elements in the four sections in the Venndiagram need to add up to 1000. university-logo Math 1300 Finite Mathematics
  44. 44. Set Properties and Set Notation Set Operations U = 1, 000 M N N ∩M university-logo Math 1300 Finite Mathematics
  45. 45. Set Properties and Set Notation Set Operations U = 1, 000 M N 300 N ∩M university-logo Math 1300 Finite Mathematics
  46. 46. Set Properties and Set Notation Set Operations U = 1, 000 M N 200 300 N ∩M university-logo Math 1300 Finite Mathematics
  47. 47. Set Properties and Set Notation Set Operations U = 1, 000 M N 200 300 300 university-logo Math 1300 Finite Mathematics
  48. 48. Set Properties and Set Notation Set Operations U = 1, 000 M N 200 300 300So, 300 commuters listen to news and not music. university-logo Math 1300 Finite Mathematics

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