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					1-6 Set Theory




                 Warm Up
                 Lesson Presentation
                 Lesson Quiz
1-6 Set Theory
 Warm Up
  Write all classifications that apply to each real
  number.
    5
 1.      rational, repeating decimal
    9
 2. 5    irrational

          rational, terminating decimal, integer,
 3. 25
          whole, natural

 4. –6    rational, terminating decimal, integer
     7    rational, terminating decimal
 5.
    10
1-6 Set Theory


    Sunshine State Standards

MA.912.D.7.1 Perform set operations such
as union and intersection, complement, and
cross product.
Also MA.912.D.7.2, MA.912.A.10.1.
1-6 Set Theory



              Objectives
Perform operations involving sets.
Use Venn diagrams to analyze sets.
1-6 Set Theory

              Vocabulary
  set
  element
  union
  intersection
  empty set
  universe
  complement
  subset
  cross product
1-6 Set Theory
 A set is a collection of items. An element is an
 item in a set. You can use set notation to represent
 a set by listing its elements between brackets. The
 set F of riddles Flore has solved is F = {1, 2, 5, 6}.
 The set L of riddles Leon has solved is L = {4, 5, 6}.
 The union of two sets is a single set of all the
 elements of the original sets. The notation F L means
 the union of sets F and L.

               Union          F L = {1, 2, 4, 5, 6}
                          3   Together, Flore and Leon
       Set F      Set L       have solved riddles 1, 2,
       1 2 56       4         4, 5, and 6.
1-6 Set Theory

 The intersection of two sets is a single set that
 contains only the elements that are common to the
 original sets. The notation F ∩ L means the
 intersection of sets F and L.


          Intersection       F ∩ L = {5, 6}
                         3   Flore and Leon have both
       set F     set L       solved riddles 5 and 6.
       1 2 56      4


 The empty set is the set containing no elements. It
 is symbolized by  or {}.
1-6 Set Theory




    Writing Math
 In set notation, the elements of a set can be
 written in any order, but numerical sets are
 usually listed from least to greatest without
 repeating any elements.
1-6 Set Theory
  Additional Example 1A: Finding the Union and
               Intersection of Sets
  Find the union and intersection of each pair of
  sets.
   A = {5, 10, 15}; B = {10, 11, 12, 13}


                 Set A   Set B
             5    15 10 11
                       12 13

   To find the union, list every element that lies in
   one set or the other.
   A U B = {5, 10, 11, 12, 13, 15}
1-6 Set Theory
       Additional Example 1A Continued

  Find the union and intersection of each pair of
  sets.
   A = {5, 10, 15}; B = {10, 11, 12, 13}


                Set A   Set B
            5    15 10 11
                      12 13

  To find the intersection, list the elements
  common to both sides.
           A ∩ B = {10}
1-6 Set Theory
  Additional Example 1B: Finding the Union and
                 Intersection
  Find the union and intersection of each pair of
  sets.
     A is the set of whole number factors of 15;
     B is the set of whole number factors of 25.
     A = {1, 3, 5, 15}            Write each set in set
     B = {1, 5, 25}               notation.
    A U B = {1, 3, 5, 15, 25} To find the union, list all of
                              the elements in either set.
    A ∩ B = {1, 5}            To find the intersection, list
                              the elements common to
                              both sets.
1-6 Set Theory
                Check It Out! Example 1a

Find the union and intersection of each pair of sets.

 A = {–2, –1, 0, 1, 2}; B = {–6, –4, –2, 0, 2, 4, 6}


 A U B = {–6, –4, –2, –1, 0, 1, 2, 4, 6}
                                  To find the union, list all of
                                  the elements in either set.

 A ∩ B = {–2, 0, 2}                   To find the intersection, list
                                      the elements common to
                                      both sets.
1-6 Set Theory
                Check It Out! Example 1b

Find the union and intersection of each pair of sets.
 A is the set of whole numbers less than 10;
 B is the set of whole numbers less than 8.
 A = {1, 2, 3, 4, 5, 6, 7, 8, 9}      Write each set in set
 B = {1, 2, 3, 4, 5, 6, 7}            notation.

  A U B = {0, 1, 2, 3,4, 5, 6,        To find the union, list all of
           7, 8, 9}                   the elements in either set.

  A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7} To find the intersection, list
                                   the elements common to
                                   both sets.
1-6 Set Theory

The universe, or universal set, for a particular
situation is the set that contains all of the elements
relating to the situation. The complement of set A in
universe U is the set of all elements in U that are not
in A.
                                    Complement of L
In the contest described on       Universe U           3
slide 6, the universe U is
                                     Set F       Set L
the set of all six riddles. The      1 2 56       4
complement of set L in
universe U is the set of all
riddles that Leon has not Complement of L = {1, 2, 3}.
solved.                      Leon has not solved riddles 1,
                             2, and 3.
1-6 Set Theory
Additional Example 2A: Finding the Complement of a
                         Set
    Find the complement of set A in universe U.
     U is the set of natural numbers less than 10;
     A is the set of whole-number factors of 9.
     A = {1, 3 ,9}; U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

     Draw a Venn diagram to     Universe U          8
     show the complement of                     2       7
     set A in universe U            Set A
                                   1    3 9             4
     Complement of A = {2,                            5 6
     4, 5, 6, 7, 8}
1-6 Set Theory

Additional Example 2B: Finding the Complement of a
                        Set

    Find the complement of set A in universe U.

    U is the set of rational numbers;
    A is the set of terminating decimals.

    Complement of A = the set of repeating
    decimals.
1-6 Set Theory




   Reading Math

  Finite sets have finitely many elements, as in
  Example 2A. Infinite sets have infinitely many
  elements, as in Example 2B.
1-6 Set Theory
              Check It Out! Example 2

 Find the complement of set A in universe U.
 U is the set of whole numbers less than 12;
 A is the set of prime numbers less than 12.

 {0, 1, 4, 6, 8, 9, 10}
1-6 Set Theory



  One set may be entirely contained within another
  set. Set B is a subset of set A if every element of
  set B is an element of set A. The notation B  A
  means that set B is a subset of set A.
1-6 Set Theory
  Additional Example 3: Determining Relationships
                 Between Sets

 A is the set of positive multiples of 3, and B is
 the set of positive multiples of 9. Determine
 whether the statement A  B is true or false.
 Use a Venn diagram to support your answer.

 Draw a Venn diagram
 to show these sets.
                                Set B
                                          Set A multiples
                                multiples of 3 that are
 False; B  A                   of 9      not multiples of
                                          9
1-6 Set Theory
             Check It Out! Example 3
 A is the set of whole-number factors of 8, and
 B is the set of whole-number factors of 12.
 Determine whether the statement A  B = B is
 true or false. Use a Venn diagram to support
 your answer.

 False; the element 8 of
 set A, is not an element       Set A     Set B
 of set B.                              1 3
                                 8
                                        42    6
                                           12
1-6 Set Theory

The cross product (or Cartesian product) of two sets
A and B, represented by A  B, is a set whose
elements are ordered pairs of the form (a, b), where
a is an element of A and b is an element of B. You
can use a chart to find A  B. Suppose A = {1, 2} and
B = {40, 50, 60}.               Set B
                            40     50     60
                    1    (1,40) (1,50) (1,60)
          Set A
                    2     (2,40) (2,50) (2,60)

 A  B = {(1, 40), (1, 50), (1, 60), (2, 40),
 (2, 50), (2, 60)}
1-6 Set Theory
         Additional Example 4: Application
 The set C = {S, M, L} represents the sizes of
 cups (small, medium, and large) sold at a
 frozen yogurt shop. The set F = {V, B, P}
 represents the available flavors (vanilla,
 banana, peach). Find the cross product C  F to
 determine all of the possible combinations of
 sizes and flavors.                     Set C
 C  F a chart to find the cross
  Make =                                S     M     L
 {(S, V), (S, B), (S, P),
  product.
 (M, V), (M,B), (M, P),            V   (S,V) (M,V) (L,V)
  Each pair represents one Set F
 (L,V), (L, B), (L, P)};           B   (S,B) (M,B) (L,B)
  combination of flavors and
 9 possible combinations               (S,P) (M,P) (L,P)
  sizes.                           P
1-6 Set Theory
               Check It Out! Example 4
   The set MN = {M, N, MN} represents the
   blood groups in the MN system. Find ABO ×
   MN to determine all of possible blood groups
   in the ABO × MN
   systems.                   M      N     MN

  Make a chart to find   A (A, M) (A, N) (A, MN)
  the cross product.     B (B, M) (B, N) (B, MN)
  Each pair represents
  one combination of    AB (AB,M) (AB,N) (AB,MN)
  ABO and MN blood
  groups.                O (O, M) (O, N) (O, M)
 ABO  MN = {(A, M), (A, N), (A, MN), (B, M),(B, N),
 (B, MN), (AB, M), (AB, N), (AB, MN), (O, M), (O, N),
 (O, MN)}: 12 possible blood groups.
1-6 Set Theory


             Lesson Quizzes


  Standard Lesson Quiz

  Lesson Quiz for Student Response Systems
1-6 Set Theory

                Lesson Quiz: Part I

 1. Find the union and intersection of sets A and B.
     A = {4, 5, 6}; B = {5, 6, 7, 8}
    A U B = {4, 5, 6, 7, 8}; A ∩ B = {5, 6}
 2. Find the complement of set C in universe U.
    U is the set of whole numbers less than 10;
    C = {0, 2, 5, 6}.
    {1, 3, 4, 7, 8, 9}
1-6 Set Theory

               Lesson Quiz: Part II

  3. D is the set of whole-number factors of 8, and E
     is the set of whole-number factors of 24.
     Determine whether the statement D  E is true
     or false. Use a Venn diagram to support your
     answer.
     true
                                      Set E        6
                           Set D
                          2    4      3
                             8   1            12
                                     24
1-6 Set Theory

              Lesson Quiz: Part III
  4. Find the cross product F  G.
     F = {–1, 0, 1}; G = {–2, 0, 2}

    F  G = {(–1, –2), (–1, 0), (–1, –2), (0, –2),
    (0, 0), (0, 2), (1, –2), (1, 0), (1, 2)}
1-6 Set Theory

   Lesson Quiz for Student Response Systems

  1. A set is defined as:

    A. a collection of items
    B. a collection of elements
    C. a union of items
    D. a union of elements
1-6 Set Theory

   Lesson Quiz for Student Response Systems

  2. The symbol  means:

     A. intersection
     B. union
     C. empty set
     D. set notation
1-6 Set Theory

   Lesson Quiz for Student Response Systems

  3. The intersection:

     A. contains common elements
     B. is the empty set
     C. contains the union
     D. contains uncommon elements
1-6 Set Theory

   Lesson Quiz for Student Response Systems

4. Find the intersection of the two sets.

  A. A  B = {1, 3, 4, 5, 6, 7}
  B. A  B = {2}                   Set A        Set B
                                    8           3
  C. A  B = {2}                  4         2        6
                                    1           12
  D. A  B = {1, 3, 4, 5, 6, 7}
1-6 Set Theory

   Lesson Quiz for Student Response Systems
 5. Find the compliment of set A in universe U.
   U = All whole-numbers less than 9
   A = All even numbers

   A. {2, 4, 6, 8}
   B. {1, 3, 6, 7, 8}
   C. {1, 3, 5, 7, 9}
   D. {1, 3, 5, 7}

				
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