Finite Math B: Chapter 7 Notes
Sets and Probability
1
Chapter 7: Sets and Probability
7.1 Sets
(Lots of new terms!)
What is a set?
A set is a __________________________ collection of objects.
We should always be able to answer the question: “Is object X in this set or not?”
Examples of “sets”
{Mrs. Leahy’s Semester 2 Discrete Math Class}
{Coins minted by the US Treasury in 2011}
{even integers}
Generally in this chapter we will be talking about sets of numbers.
Set Notation:
Use set braces: { } to enclose the numbers in set.
Example 1:
1,3,5,7,9
The numbers 1, 3, 5, 7, 9 are called the __________________ or _________________ of the set.
Naming Sets:
Use the symbol: 
3 1,3,5,7,9
SAY: “3 is an element of the set”
Use the symbol: 
4 1,3,5,7,9
SAY: “4 is NOT an element of the set”
Sets are often named with letters:
Example 2:
True or False?
A  10, 20,30
A  10, 20,30
11  A
An empty set is a set with __________________________.
25  A
10  A
Symbol: _____
Can you think of a “collection” or group that would have no members?
The universal set is a set that contains all the objects being discussed. (integers, people, whole numbers, etc.)
Finite Math B: Chapter 7 Notes
Sets and Probability

2
0
= the number zero
0
= a set that has one element – the number zero

= a set that has one element – the empty set
CAUTION:
Example 3:
= an empty set, a set with no elements
How many elements are in each set?
Symbol: n ( A) = the number of elements in a finite set A.
a. A  5,6,7
b. B  {positive integers less than 5}
c.
C  0
Two sets are equal if they contain ______________ the ____________ elements.
Example 4:
True or False?
NOTE:
a. {1,10,100}  {100,1,10}
If Set A equals Set B, we say
AB
b. {32,33,34}  {32,33,34,35}
c.
d.
positive even numbers
 8  {0, 2, 4,6,8}
If Set A does not equal Set B,
we say:
3,5,7,9  5,7,9,3
Set-Builder Notation:
Useful when we are looking for objects that share a common property
 x x has property P
SAY: “The set of all elements x such that x has property P”
Example 5:
 x x is a whole number between 7 and 10
Elements: _________________________
*****
A B
Finite Math B: Chapter 7 Notes
Sets and Probability
3
Subsets
Consider the two sets A and B:
A  {3, 4,5, 6}
B  {2,3, 4,5, 6, 7,8}
Notice that EVERY element is A is also an element in B.
We say that A is a subset of B.
Example 6:
A  {1, 2,3, 4}
B  {1, 2,3, 4,5}
True or False:
Example 7:
A  {5, 6, 7,8}
B  {7, 6,5,8}
True or False:
A B
A B
Is every element in A an element in B?
A B
A B
Are A and B the same or different?
Is A a proper subset?
B A
Is every element of B an element in A?
B A
Example 8:
B A
B A
A  {1, 2,3, 4, 7}
B  {1, 2,3, 4,5,12}
True or False:
A B
A B
BØ A
The empty set is a subset of every set.
A set is always a subset of itself.
Finite Math B: Chapter 7 Notes
Sets and Probability
4
Counting Subsets
Example 9: How many subsets are possible? What are they?
(Tree diagrams can sometimes be useful)
a.
5, 6
b.
x, y, z
Basically, there are two different possibilities for each element = yes or no
SO:
Example 10: Find the number of subsets for each set.
a.
{9,8, 7, 6}
b.
 x x is a day of the week
c. 
Finite Math B: Chapter 7 Notes
Sets and Probability
5
Venn Diagrams:
Helps to show elements of sets & subsets.
U is the universal set
A is a subset of B.
B is a subset of U.
Example:
Set Operators:
Often we will form new sets by combining or manipulating one or more existing sets.
Complement:
The elements in the universal set NOT in
your set.
A (white) is a set.
A’ (pink) --- say “A prime” is everything
else and is called the complement of A.
For Example:
Let U = the students in a class
A = the set of all female students in the
class.
A’ = the set of all male students in the
class.
Example 11: Let U  {1, 2,3, 4,5, 6, 7,8,9,10} , A  1, 2,3, 4 , and B  {1,3,5, 7,9}
Find each set.
a.
A'
c.  '
b.
B'
d.
 A ' '
Finite Math B: Chapter 7 Notes
Sets and Probability
6
Intersection:
The intersection of two sets is the set of all elements
belonging to BOTH A and B.
A B
Symbol:
Example:
Let
U = the students in a class
A = the set of all male students
B = the set of all students with “B”
averages.
Then
A  B = the set of all male students with “B” averages.
Example 11: Let A  1, 2,3 , B  {1,3,5, 7} , and the universal set U  {1, 2,3, 4,5, 6, 7,8}
Find each set:
a.
A B
Disjoint Sets:
Disjoint sets have no elements in common.
The intersection of disjoint sets is the empty set.
Example:
{1, 2,3, 4}  {5, 6, 7,8}  
Union:
The set of all elements belonging to set A, to
set B, or to both sets.
Symbol: A  B
Example:
Let U = students in a class
A = set of all female students
B = set of all students with B averages
A  B = any student who is either
female OR has a “B” average
b.
A ' B
Finite Math B: Chapter 7 Notes
Sets and Probability
7
Let A  {1, 2,3, 4,5} , B  {1,3,5, 7,9} , and C  {1, 2,5, 6,9,10}
for the universal set U  {1, 2,3,....,11,12}
Example 12:
Find each set:
a.
A B
b.
AC '
c.
 A  B  C
d.
( A ' B)  C '
*****
Applications of Sets
Example:
Pg 348
The following table give the 52-week high and
low prices, the closing price, and the change
from the previous day for six stocks in the
Standard & Poor’s 100 on April 11, 2006.
Let the universal set U consist of the six stocks
listed in the table. Let A contain all stocks with
a high price greater than $34. Let B contain all
stocks with a closing price between $26 and
$30. Let C contain all stocks with a positive
price change.
State the elements in each set:
A=
A’ =
B=
AC =
C=
A B =
Finite Math B: Chapter 7 Notes
Sets and Probability
8
Example:
A department store classifies credit applicants by gender, marital status, and employment status. Let the
universal set be the set of all applicants, M be the set of all male applicants, S be the set of single applicants, and
E be the set of employed applicants. Describe each set in words.
a.
M E
b.
M ' S
c.
M ' S '
d.
M E'
7.2 Applications of Venn Diagrams
pg 353 “The responses to a survey of 100 household show that 21 have a DVD player, 56 have a videocassette
recorder, and 12 have both. How many have neither a DVD player nor a videocassette recorder?”
Venn Diagrams are very useful for “sorting out” this information.
Things that might happen:
1 set leads to 2 regions
2 sets lead to 4 regions
2 sets lead to 3 regions
2 sets lead to 3 regions
3 sets lead to 8 regions
Finite Math B: Chapter 7 Notes
Sets and Probability
9
Example 1: Shade the following Venn Diagrams
a.
A B
A B
b.
A
C
d.
A ' B '
A
B
A ' C '
A
B
C
A ' ( B  C ')
e.
B
C
B
C
f.
B  AC
A
A
C
c.
A
B
C
B
Example 2:
“The responses to a survey of 100 household show that
21 have a DVD player, 56 have a videocassette recorder,
and 12 have both. How many have neither a DVD player
nor a videocassette recorder?”
In general: These questions are often referred to as Both/Neither Questions and usually have information
about two types of elements: Type A and Type B.
Total = Type A + Type B + Neither – Both
Finite Math B: Chapter 7 Notes
Sets and Probability
10
Example 3: A survey of 77 freshman business students at a large university produced the following results.
25 read Business Week
19 read The Wall Street Journal
27 do not read Fortune
11 read Business Week but not The Wall Street Journal
11 read The Wall Street Journal and Fortune
13 read Business Week and Fortune
9 read all three
Questions to answer:
How many read none of the publications?
How many read only Fortune?
How many read Business Week and The
Wall Street Journal but not Fortune?
The Union Rule For Sets:
Essentially:
x  y  z  n(A) + n(B)  y
Example 4: A group of 10 students are all majoring in either accounting or economics or both. Five of the
students are economics majors and 7 are majors in accounting. How many major in both subjects?
Finite Math B: Chapter 7 Notes
Sets and Probability
11
Example 5:
The table gives the number
of threatened and
endangered animal species
in the world as of April
2006. (pg 358) Using the
letters given in the table to
denote each set, find the
number of species in each
of the following sets.
a.
EB
b.
EB
c.
(F  M )  T '
Example 6: Suppose that a group of 150 Students have joined at least one of three chat rooms: one on autoracing, one on bicycling, and one for college students. For simplicity, we will call these rooms A, B, and C. In
addition,
90 students joined room A
50 students joined room B
70 students joined room C
15 students joined rooms A and C
12 students joined rooms B and C
10 students joined all three rooms
How many students joined both A and B?
Finite Math B: Chapter 7 Notes
Sets and Probability
12
Example 7: A survey was conducted of 150 High School students who were asked about which international city
they would like to visit from Athens, Dublin, and Hong Kong. The results were tallied as follows:
60 students wanted to visit Athens
70 students wanted to visit Dublin
80 students wanted to visit Hong Kong
10 students did not respond to the survey
25 wanted to visit both Athens and Dublin
15 wanted to visit both Athens and Hong Kong
35 wanted to visit both Dublin and Hong Kong
How many students responded that they wanted to visit all three cities?