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Chapter 2

Chapter 2. Set Theory. 2.1 Basic Set Concepts. Information obtained from surveys must be sorted and organized in order to analyze it. Set theory arises from the need to find this order and classify things into collections. The collections are sets. Definitions.

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Chapter 2

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  1. Chapter 2 Set Theory

  2. 2.1 Basic Set Concepts Information obtained from surveys must be sorted and organized in order to analyze it. Set theory arises from the need to find this order and classify things into collections. The collections are sets.

  3. Definitions • Set – collection of objects whose contents can be clearly determined or are well defined A set is usually denoted by a capital letter. • Element – an object in a set Example If T is the set of teachers at BHS, then Ms. Collier is an element of that set T.

  4. Definitions • Set – collection of objects whose contents can be clearly determined or are well defined Determine if each of the following is a set: • Actors who won an Oscar • Great actors • Best movie theatre snacks • BHS students with 3.0 or higher GPAs YES NO NO – not well defined; who decides? NO YES

  5. Number Sets • Natural numbers – counting numbers N= { 1, 2, 3, 4, … } Listing objects in { } is a roster of the set. • Whole numbers – naturals plus zero W= { 0, 1, 2, 3, 4, … } • Integers – wholes plus opposites Z= { …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }

  6. Three ways to designate a set Word description and set-builder are practically same. • Roster Method • Word Description • Set-Builder Notation W = { Saturday, Sunday } W is the set of days in the weekend W = { x | W = { x | x is a day in the weekend }

  7. Three ways to designate a set • Roster Method • Word Description • Set-Builder Notation C = { penny, nickel, dime, quarter, half-dollar } C is the set of U.S. coins with value less than $1 C = { x | x is a U.S. coin with value < $1 }

  8. Three ways to designate a set Exclusive – does not include the endpoints • Roster Method • Word Description • Set-Builder Notation P = { 1, 2, 3, 4 } P is the set of whole numbers between 0 and 5 exclusive ^ P = { x | x is a whole number and 0 < x < 5 }

  9. Three ways to designate a set Inclusive –includes the endpoints • Roster Method • Word Description • Set-Builder Notation P = { 0, 1, 2, 3, 4, 5 } P is the set of whole numbers between 0 and 5 inclusive ^ P = { x | x is a whole number and 0 < x < 5 }

  10. But absolutely do NOT use both symbols like this: {  } Notations • Empty set • set containing no elements • denoted by { } or • Element of • one of the objects contained in the set • denoted • “not an element of” is denoted

  11. True or False? True True True False True False True

  12. Definition • Cardinal number – the number of elements in a set; also called the cardinality and denoted n(A) for the cardinality of set A Example V = { a, e, i, o, u } Find n(V). = 5 Example Q = { 1, 3, 5, 7, 9, 11, 13} Find n(Q). = 7 Example L = { x | x is a letter in the alphabet } Find n(L). = 26

  13. Definition • Cardinal number – the number of elements in a set; also called the cardinality and denoted n(A) for the cardinality of set A You can find the cardinality of a finite set. Otherwise, the set is infinite. Example B = { 0 } Find n(B). = 1 Example Z = Ø Find n(A). = 0 Example M = { 2, 4, 6, … } M is an infinite set.

  14. Equal sets are always equivalent, BUT equivalent sets are not always equal! Definitions • Equal sets – sets with exactly the same elements (regardless of order or repetition) • Equivalent sets – sets with the same cardinal number (same number of objects) Example A = { w, x, y, z } B = { z, y, w, x } Are A and B equivalent sets? Equal sets? They are equal AND equivalent. Example C = { w, x, y } D = { a, b, c } Are A and B equivalent sets? Equal sets? They are equivalentbut NOT equal.

  15. 2.2 Venn Diagrams & Subsets Venn Diagrams are the visual representation of sets. We will examine the components of Venn diagrams and how to interpret them.

  16. Definition • Universal set – the set that contains all the elements being considered in a given problem, denoted U Example A = {Wheel of Fortune, Minute to Win It} B = {Jeopardy!, Cash Cab, Family Feud} U = ? Game Shows

  17. John Venn • British logician and philosopher • Wrote three books on logic, including Symbolic Logic in 1881 • Created Venn diagrams to represent sets

  18. Definition • Universal set – the set that contains all the elements being considered in a given problem, denoted U • In Venn diagrams, the universal set is denoted by a rectangle around the entire problems. • Sets are denoted with ovals or circles in diagram.

  19. Definition • Complement – the set of all elements in the universal set that are NOT in the given set, denoted A ’ for the complement of A Example A = {1, 3, 5, 7, 9} U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A’ = ? {2,4,6,8}

  20. Definition Do not get  and  confused!  is for elements  is for subsets • Subset – set A is a subset of B if every element in A is also in B; denoted or • The symbol denotes a proper subset; all subsets except the set itself are proper subsets “contained in” “contained in or equal to” Example Consider the following statements: 3  { 1, 2, 3, 4, 5 } { 3 }  { 1, 2, 3, 4, 5 } 3  { 1, 2, 3, 4, 5 } { 3 }  { 1, 2, 3, 4, 5 } The first and second are true; the last ones are false. Example A = {1, 3, 5, 7, 9} B = {2, 4, 6} C = {3, 5} D = { 2, 4, 6} U = {1, 2, 3, 4, 5, 6, 7, 8, 9} Example A = {1, 3, 5, 7, 9} B = {2, 4, 6} C = {3, 5} D = { 2, 4, 6} U = {1, 2, 3, 4, 5, 6, 7, 8, 9} C A C A B A B D

  21. Subsets • Every set has the empty set as a subset • Every set has the set itself as a subset Example M = {3, 5, 7} List all the subsets. Example M = {3, 5, 7} List all the subsets. Ø {3, 5, 7} Example M = {3, 5, 7} List all the subsets. Ø {3} {3, 5, 7} {5} {7} Example M = {3, 5, 7} List all the subsets. Ø {3} {3, 5} {3, 5, 7} {5} {3, 7} {7} {5, 7} The number of subsets is equal to 2n, where n is the set’s cardinality.

  22. There should be 24 = 16 subsets! Subsets • Every set has the empty set as a subset • Every set has the set itself as a subset Example M = {a, b, c, d} List all the subsets. Example M = {a, b, c, d} List all the subsets. Ø {a, b, c, d} Example M = {a, b, c, d} List all the subsets. Ø {a} {a, b, c, d} {b} {c} {d} Example M = {a, b, c, d} List all the subsets. Ø {a} {a, b} {a, b, c, d} {b} {a, c} {c} {a, d} {d} {b, c} {b, d} {c, d} Example M = {a, b, c, d} List all the subsets. Ø {a} {a, b} {a, b, c} {a, b, c, d} {b} {a, c} {a, b, d} {c} {a, d} {a, c, d} {d} {b, c} {b, c, d} {b, d} {c, d}

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