# 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
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
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 }.
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
must have all the
A                         B
AB-            antigens found in
A-               B-
the donor's blood.
AB+
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+
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
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
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
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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