Thinking Mathematically by Robert Blitzer by xumiaomaio

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									  Thinking
Mathematically
     Chapter 2
    Set Theory
          Basic Set Concepts
A set is a collection of objects. Each object is
called an element, or member of the set.
Often the objects in a set are listed and are
enclosed in “braces.”
For example the set of integers that fall
between 1 and 5 can be written {2 , 3 , 4}.
                  Representing Sets
    • Word Description: Describe the set in
      your own words, but be specific so the
      elements are clearly defined
         All the whole numbers from 1 to 20
    • Roster Method: List each element,
      separated by commas, in braces
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, .12, 13, 14, 15, 16, 17, 18, 19, 20}
                          {1, 2, 3, . ., 20}
    • Set-Builder Notation:
          {x | x is … word description}
    {x | x is a whole number and 1 ≤ x ≤ 20}
           The Empty Set
The empty set, also called the null set, is
the set that contains no elements.
The empty set is represented by
                     {}
or by
                     
We will use { } most of the time, because
it's easier to understand.
            Elements of a set
• The symbol  is used to indicate that an object
  is an element of a set. The symbol is used to
  replace the words:
      is an element of, or belongs to
• The symbol  is used to indicate that an object
  is not an element of a set. The symbol is used to
  replace the words:
      is not an element of, or does not belong to
          Elements of a set

• 8{2,4,6,8}
• you{x | x is a student in this class}
• Dr. Landis  {x | x is a student in
                        this class}
• f  { a, b, c, d, e, g, h, i, k , l, m}
     Counting and the Natural
            Numbers
• Sets are collections of elements.
  Sometimes we want to count how many
  elements a set has.
• The natural numbers are the "counting
  numbers":
     N = {0, 1, 2, 3, 4, 5,…}
Definition of a Set’s Cardinal Number


 The cardinal number of set A, represented
 by n(A), is the number of elements in set A.
 The symbol n(A) is read “n of A”.
            Cardinal Number
If A ={a, b, c}, what is n(A)? n(A) = 3
    0 1    2 3
If A = {}, what is n(A)? n(A) = 0
     0
If A = {a, b, c, d, a, e}, what is n(A)? n(A) = 5
     0 1   2 3 4 5 no
                 4 5   element can be counted
                    twice, even if it's
                    accidentally listed twice!!!!!
  Definition of a Finite Set

Set A is a finite set if n(A) is a natural
number. A set that is not finite is called
an infinite set.
The set of natural numbers, for example,
is itself an infinite set.
Definition of Equality of Sets
Set A is equal to set B means that set A
and set B contain exactly the same
elements, regardless of order. We
symbolize the equality of sets A and B
using the statement A = B.
       Examples: Equal Sets
• Working left to right, cross out every
  occurrence of the given element. If
  anything is left over, the sets are not
  equal.
• {a, b, c}  {b, a, c}

• {a, b, c, d}  {a, b, c, d, e}

• {a, b, c, d}{a, c, d, d, b, a, b}
  Definition of Equivalent Sets
  Set A is equivalent to set B means that set A
  and set B contain the same number of
  elements. For equivalent sets, n(A) = n(B).
  The sets
{G. Washington, J. Adams, T. Jefferson, J. Madison}
        and
{1789, 1797, 1801, 1809}
  are equivalent because they both have a
  cardinality of 4.
 Definition of Equality of Sets
• Remember:
 – Equal means the sets are exactly the same.
 – Equivalent means that the sets have the
   same number of elements.
 – Every semester, lots of students lose points
   on an exam because they forget this!!
              Section Quiz
• Problem 1: Represent the set {x | 1 < x ≤ 9}
  using the roster method. {2, 3, 4, 5, 6, 7, 8, 9}
• Problem 2: Are the sets {a, b, c} and
  {c, a, b, a} equivalent?
           yes. they both have the cardinality 3
• Problem 3: Are the sets {a, b, c} and
  {c, a, b, a} equal? yes. they contain exactly the
                       same elements
  Thinking
Mathematically
     Section 2
      Subsets
  Definition of a Universal Set
• When we talk about a set, we ask what's
  in the set and what's not in the set.
• Well, pretty much anything you can
  think of might not be in the set.
• We limit ourselves to what makes sense.
• The universal set is one that contains all
  of the elements that are included in the
  discussion.
  Definition of a Universal Set
• Examples:
  – When talking about the set {a, b, c, e} our
    universe most likely would be the set of
    English language lowercase letters.
  – When talking about students in this
    classroom, my universe might be all FDU
    students taking Comprehensive Math. It
    might also be all students at FDU. It might
    be all people in the U.S.
Definition of the Complement of a Set
 The complement of a set is the collection
 of all the objects in the universal set that
 are not in the given set.
 The complement of a set A is written A´.
       A´= {x | x  U and x  A }.
  I have made those apostrophes blink to
  emphasize that you should always watch
  out for them, especially on exams.
                Examples
• Suppose I said "Consider the set of students
  who are actually listening to me now."
• The universal set, or universe under
  discussion, would be the set of all students
  in this classroom. I'm not interested in
  chairs, tables, books, or even students in
  other classrooms.
• The complement of the set would be anyone
  not listening to me.
   Definition of a Subset of a Set

Set A is a subset of set B, expressed as
                   AB
if every element in set A is also in set B.

Note that the set A could be equal to the
set B. That's why there's a line at the
bottom of the symbol. Think about how ≤
means less than or equal.
Definition of a Proper Subset of a Set

Set A is a proper subset of set B,
  expressed as A  B, if set A is a subset
  of set B and sets A and B are not equal
  ( A  B ).
Note that the set A can not be equal to
the set B. That's why there isn't a
line at the bottom of the symbol.
 The Empty Set as a Subset

1. For any set B, { }  B.

2. For any set B other than the empty
   set,             { }  B.

3. Of course, { } might also be
   written as .
             Number of subsets
•    How many subsets does {a, b, c} have?
•    Let's count:
                     choose a   choose b   choose c
    1.   {}            no         no        no
    2.   {a }          yes        no        no
    3.   {b }          no         yes       no
    4.   {c}           no         no         yes
    5.   {a, b}        yes        yes       no
    6.   {a, c}        yes        no        yes
    7.   {b, c}        no         yes       yes
    8.   {a, b, c}     yes        yes       yes
         Number of subsets
• For each element of the set, we could either
  choose or not choose that element.
• Every set of different choices forms a
  different subset.
• Since there are two choices for a, two
  choices for b, and two choices for c, there
  are 2  2  2 choices.
• 2  2  2 = 23 = 8
   Number of Subsets and Proper
             Subsets
• The number of subsets of any set is given
  by: 2n
    2n means 2 x 2 x 2 x . . . x 2, n times.


  The number of proper subsets of any set
  is given by: 2n - 1
              Section Quiz
• Problem 1: If the universe is {1, 2, 3, 4, 5,
  6, 7, 8} and A = {1, 3, 5, 7}, what is A' ?
• Problem 2: True or false? {2, 4, 6, 8}
   – {}  {a, b, c}       true
   – {}  {a, b, c}       false
   – {b}  {a, b, c}      true
  Thinking
Mathematically
        Section 3
  Venn Diagrams and Set
       Operations
           Venn Diagrams
                         U
Disjoint sets have no        A       B
elements in common.
                         U       A
The set B is a proper
                                 B
subset of A.
                         U
 The sets A and B have       A       B
 some common elements.
           Venn Diagrams
• A general Venn Diagram looks like the
  one below, with the understanding that
  the purple center region might be empty
  or that one set might be inside the other.


             A       B
                                Venn Diagrams
• Consider the case the universe of {1, 2, ..., 8}, with
  A = {1,2, 3, 4} and B = {2, 4, 6, 8}. Let's see how
  these values get placed in the Venn Diagram
  below:
                              but 2 is also B
                            but 4 is also in in B!
1
    2
        3
            4
                5                       A       B
                    6
                        7
                            8
          Venn Diagrams
• The area representing those elements of
  A that don't belong to B is the region:



               A       B
          Venn Diagrams
• The area representing those elements
  that are both in A and in B is:



               A      B
          Venn Diagrams
• The area representing those elements of
  B that don't belong to A is the region:



               A       B
           Venn Diagrams
• The area representing those elements
  that don't belong to either A or B is:



                A       B
 Definition of Intersection of
             Sets
The intersection of sets A and B, written
               AB
is the set of elements common to both set
A and set B. This definition can be
expressed in set builder notation as
follows:
    A  B = { x | x  A AND x  B}
 Definition of Intersection of
             Sets
The intersection of sets A and B
  A  B = { x | x  A AND x  B}


             A         B
  Definition of Union of Sets

The union of sets A and B, written
               AB
is the set of elements that are members of
set A or of set B or of both sets. This
definition can be expressed in set builder
notation as follows:
    A  B = { x | x  A OR x  B}
  Definition of Union of Sets
The union of sets A and B, written
  A  B = { x | x  A OR x  B}



              A          B
         DeMorgan's Laws
• Remember what happened when we
  considered those don't belong to A or to B?
  That's the complement of A or B, namely
                (A  B)'
         DeMorgan's Laws
     The the region is  sae
  ••The purple region therepresents B'
 Therefore A'redB' =below B)' represents A'
     Now,blue region (A below as what
 If we combine these Venn Diagrams,
    was left blue = purple, slide:
since red + in our previousthe purple
region represents A'  B'


   A'                                B'
        DeMorgan's Laws
• DeMorgan's Laws state that

         (A  B)' = A'  B'
 and

         (A  B)' = A'  B'
                 Section Quiz
• Problem 1: Given the Venn Diagram
             8                     1

                     4
                     A   2
                          5
                              B        6
                 3
                              7



  –   Describe set A in roster notation. {2, 3, 4, 5}
  –   Describe set A' in roster notation. {1, 6, 7, 8}
  –   Describe A  B' in roster notation {3, 4}
  –   Describe A'  B in roster notation. {1, 2, 5, 6, 7, 8}
  Thinking
Mathematically
         Section 4
  Set Operations and Venn
  Diagrams with Three Sets
Venn Diagrams - Two Sets
           IV: In not B,  A,
     Region III:A A of B, (A  B)' A', B
The RegionregionsBandtheAB, A BDiagram
    four I:II:NotbutA ornotVennB', A - B - A
     Region InIn in but in in B




             A           B
             I     II     III     IV
Venn Diagrams - Three Sets
      Region VII: In Aand CC VennCC
      Region III: V: Inand not B B or Diagram
           regionsC but not in A
The eightRegionIn ABof thebutandorCBA
     Region IV: VIII: butand A, notorin C
       RegionVI: In B Not Bbut B in B
        RegionII:
               I:
         Region         A in in not
                                  A


              A                    B
                I        II    III VIII
                         V
                    IV        VI
                         VII

                         C
     Venn Diagrams - Three Sets

Region I: In A but not in B or C
                                     U
Region II: In A and B but not in C       A                    B
Region III: In B but not in A or C       I        II         III
Region IV: In A and C but not in B                V
                                             IV         VI
Region V: In A and B and C
                                                  VII              VIII
Region VI: In B and C but not in A
                                                        C
Region VII: In C but not in A or B
Region VIII: Not in A, B or C
   Example: Blood Typing
 Blood is characterized by examining
  components called antigens.
 Two of these antigens are called type A and
  type B.
 We name a person's blood on whether or
  not they have these antigens:
 A: has only antigen type A
 B: has only antigen type B
 AB: has both
 O: has neither.
   Example: Blood Typing

Look at the diagram below:

                             has only A

                             has only B
          A           B
               AB            has both A
                             and B
    O                        has neither
                             A nor B
   Example: Blood Typing
 But there's a third antigen as well: the Rh
  antigen.
 Blood with Rh is said to be positive: +
 Blood without Rh is said to be negative: -
   Example: Blood Typing
Look at the diagram below:

           -
           A                 B
               A AB- B
                 AB     B-
           A-
                 AB+ B+

          O
              A+
                   +
                   O+
          O-            Rh
 Example: Blood Typing
                              When you receive
                              blood, your blood
                              must have all the
A                         B
               AB-            antigens found in
     A-               B-
                              the donor's blood.
               AB+
          A+         B+       Who can receive
                O+            ANY type of blood?
O-                   Rh
                               Universal
                               recipient: AB+
 Example: Blood Typing
                              When you donate
                              blood, the
                          B
                              acceptor's blood
A              AB-            must have all the
     A-                B-
                              antigens found in
               AB+
                     B+       your blood.
          A+
                O+            Who can donate to
O-                Rh          everyone?
                              Universal
                              donor: O-
                 Section Quiz
• Problem: Suppose the Venn diagram below
  represented A: people who like candy, B: people
  who like soda, and C: people who like liver.
   – what region(s) represent people who like all three?

                   AA              B              V
                      I
                             II  III   VIII
                          IV V VI
                            VII

                             C
                 Section Quiz
• Problem: Suppose the Venn diagram below
  represented A: people who like candy, B: people
  who like soda, and C: people who like liver.
   – what region(s) represent people who like only one of
     the three?

                   AA              B             I, III
                      I
                             II  III   VIII
                          IV V VI
                                                 and VII
                            VII

                             C
                 Section Quiz
• Problem: Suppose the Venn diagram below
  represented A: people who like candy, B: people
  who like soda, and C: people who like liver.
   – what region(s) represent people who like candy, but not
     soda?
                   AA              B             I and
                      I
                             II  III   VIII
                          IV V VI
                                                 IV
                            VII

                             C
  Thinking
Mathematically
           Section 5:
 Surveys and Cardinal Numbers
Cardinal Number of the Union
         of Two Sets
• Suppose a class has 16 students with brown hair and
  that it has 12 students who wear glasses.
• How many students in the class either have brown
  hair or wear glasses?
• Since a student in this group can either have brown
  hair (16 students) or wear glasses (12 students) a
  good guess is that there are 16 + 12 = 28 students in
  this group.
• Let's count!
Blink for wearing wear glassesleave have
          I haven't mentioned, for
Everyoneme if youglasses, blink ANDthe
Everyone with brown hair, blink for me.
room.
me. hair.
brown

       11
        1     2
              2                   3
                                  2

       4
       3             3
                     5
        6
        4            7
                     5             8
                                   6

 74
 2
 9     5
       10     11
              36
              8             9
                           12
7
13     10
       14     15
              8      16
                      9     11
                            17    18
                                  10 +
                                      -
19
12     11
        4
       13
       20     21
              14     22
                     15            23
                                   5
                                  16 28
                                  12    23
Cardinal Number of the Union
         of Two Sets
• What happened there?
• When counting heads, we counted the
  intersection twice, first as having brown
  hair and then again as wearing glasses.
• We have to take that into account.
• We subtract that number in the
  intersection from the total.
Cardinal Number of the Union
         of Two Sets
• The number of students who have brown
  hair or who wear glasses is the UNION of
  two sets. (Remember, or means union.)
• The number of students who have brown
  hair and who wear glasses is the
  INTERSECTION of two sets.
  (Remember, and means intersection.)
 Cardinal Number of the Union of
16 = 5 + 11
            Two Sets
                             12 = 5 + 7
brown hair                                 glasses

                      5
             16                       12
      11                                    7



n(brown hair) = 16           n(glasses) = 12
       n(brown hair  glasses) = 5
n(brown hair  glasses) = 11 + 5 + 7 = 23
   Formula for the Cardinal
  Number of the Union of Two
             Sets
n(A  B) = n(A) + n(B) - n(A  B)
To find the cardinal number in the union of
 sets A and B, add the number of elements
 in sets A and B and then subtract the
 number of elements common to both sets.
   Formula for the Cardinal
 Number of the Union of Three
             Sets
n(A  B  C) = n(A) + n(B) + n(C)
      - n(A  B) - n(A  C) - n(B  C)
      + n(A  B  C)
This one is tougher, and you don't have to
memorize it if you don't want to. You'll see
we can solve problems without it.
     Solving Survey Problems
1. Use the survey’s description to define sets and
   draw a Venn diagram.
2. Use the survey’s results to determine the
   cardinality for each region in the Venn
   diagram. Start with the intersection of the
   sets, the innermost region, and work
   outward.
3. Use the completed Venn diagram to answer
   the problem’s questions.
       Solving Survey Problems
1. It's easier if we do an example.
2. Sixty people were contacted and responded to
   a movie survey. The following results were
   obtained.
  1.   6 people liked comedies, dramas AND sci-fi.
  2.   13 people liked comedies and dramas.
  3.   10 people liked comedies and sci-fi.
  4.   11 people liked dramas and sci-fi.
  5.   26 people liked comedies.
  6.   21 people liked dramas.
  7.   25 people liked sci-fi.
workin the center:
start finally..... out.
continue further
 and outwards:
Step 1: Draw a Venn diagram.
                              Survey
  Sixtypeople liked comediesand sci-fi.
  21 11 people were contacted and responded
 25people liked comedies, dramas AND sci-fi.to a movie survey.
  26 people liked comedies and dramas.
                  comedies.
                  dramas.
  13people liked sci-fi.
  6 10 people liked dramas and sci-fi.
                                                these two numbers
                                                must add up to 13
                                                                        9
          comedy                                drama
                                                                        7
                          9       7                                    7
                                                                       43
                      9           7         3
                                            3
                                                                       6
                                                                       64
                                                                     +5
                                                                     +4 6
                                  66                                 +5
                              4
                              4        55                            ----
                                                                        5
                                                                     ----
                                                                     18
                                                                     17
                                                                    +10
                                                                     15
                                  10             16                 ----
                                  10                                   21
                                                                       26
                                                                       44
                 sci fi                                                25
                                                                     -18
                                                                     -17
                                                        U            -15
                                                                     ----
                                                                      60
                                                                     ----
                                                                    -449 3
                                                                       10
                                                                    ----
  Solving Survey Problems
       1.   6 people liked comedies, dramas AND sci-fi.
       2.   13 people liked comedies and dramas.
       3.   10 people liked comedies and sci-fi.
       4.   11 people liked dramas and sci-fi.
       5.   26 people liked comedies.
       6.   21 people liked dramas.
       7.   25 people liked sci-fi.           How many
                                               How many
                                              people liked
                                               people don't
                                               like one type
                                              only movies atof
U 16         C       C      D                 movie?
              9     7    3                     all?
                     65                         9 + 3 + 10 =
                 4                                16
                                                22
                    10
                    SF
 Solving Survey Problems
A class has 28 students. Of the 15
female students, 8 wear glasses. Half
the class (14 students) wear glasses.
How many students are either male or
don’t wear glasses?
                Venn Diagrams
                                            How many
                     Females with glasses   students are
         glasses
  Females, no
                                            either male or
                                            don’t wear
                                            glasses?
                Female15 14 Glasses
                       8
                   7      6
     glasses    7
                                            
                               Males with glasses
Males, no
                 28 students        7+7+6=20
               15 female students
             14 wear glasses
        8 female students wear glasses
      Or Use DeMorgan's Law
•   Male = not Female = F '
•   don’t wear glasses = G '
•   F '  G' = (F  G)'
•   n(F  G) = 8
•   n((F  G)') = 28 – 8 = 20
                Section Quiz
 50 students were contacted and responded to a school survey.
 The following results were obtained.
    1. 3 students liked math, history and literature.
    2. 5 people liked math and literature.
    3. 15 people liked history and literature.
    4. 6 people liked history and math.
    5. 30 people liked history.
    6. 8 people liked math.
    7. 31 people liked literature.
A. How many students like only math?
B. How many students like either history or literature,
   but not math?
C. How many students don't like any of these subjects?
                    Section Quiz
50 students were contacted and responded to a school survey. The
      following results were obtained.
     1. 3 students liked math, history and literature.
     2. 5 people liked math and literature.
     3. 15 people liked history and literature.
     4. 6 people liked history and math.
     5. 30 people liked history.
     6. 8 people liked math.
     7. 31 people liked literature..


                          U 4         M
                                               3
                                                        H
0                                      0              12
                                               3 12
                                           2
38
                                               14
4                                              L
                      Section Quiz
• During a survey of 100 people who were asked to name
  their three favorite flavors of ice cream, the following was
  noted.
   –   15 people liked vanilla, chocolate and strawberry.
   –   35 people liked vanilla and chocolate
   –   27 people liked vanilla and strawberry
   –   25 people liked chocolate and strawberry
   –   60 people liked vanilla
   –   70 people liked chocolate
   –   40 people liked strawberry
• How many people liked only vanilla?
                           Section Quiz
•   During a survey of 100 people who were asked to name their three favorite
    flavors of ice cream, the following was noted.
     –   15 people liked vanilla, chocolate and strawberry.
     –   35 people liked vanilla and chocolate
     –   27 people liked vanilla and strawberry
     –   25 people liked chocolate and strawberry
     –   60 people liked vanilla
     –   70 people liked chocolate
     –   40 people liked strawberry                    strawberry                       vanilla

•   How many people liked only vanilla?                                  12        13
                                                                         15
                                                                    10        20
                                      13
                                                         chocolate

								
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