Finite Mathematics Lesson 5
Sections 5.1  5.3
Section 5.1 Sets
This section will be a nice break from matrices. This section explains things well so I will just list the important concepts.
TERM  DEFINITION  EXAMPLE 

collection of  A = {1,2,b} set A contains 3 elements 

all elements are contained in another set  A = {1, 2, b} B = {1, b } B is a subset of A 

set containing all elements  U ={ all fifty states} 

elements not in a given set (A' is said A prime)  If U ={ 1,2,3,4,5} and A = {1,2,3} then A' = {4,5} 

set containing no elements  sometimes called the null set �  � is a subset of all sets 

set containing elements that belong to both set A and set B,  If A = {1, 3, 5} and B ={3, 4, 6 } , then = {3} 

set containing elements that belong to both set A or set B,  If A = {1, 3, 5} and B ={3, 4, 6 } , then = {1, 3, 4, 5, 6} 
When you are ready, click here for the assignment for this section.
We are going in a new direction now that we are in chapter 5. Don't forget all that linear programming stuff. It will be on the midterm. A sample midterm with solutions will be posted.
Counting
Counting is a general term that describes a
method for counting objects in a set. We may want to count the number of objects in
a set or the numbers of ways the objects can be arranged. We could also count the
number of possible poker hands that could be dealt given a single deck of cards.
Many of these type of counting problems are much to difficult to use a trail and error
approach. Could you image actually trying to count out each possible hand of
poker?. Chapter 5 will explore different counting techniques.
Course Notes
Section 5.2 A Fundamental Principle of Counting
Read the section first.
InclusionExclusion Principle
When counting objects that belong to more than
one set, we want to avoid counting objects more than once. Lets say set A
contains all the students in this course that are male, set B
contains all the student that are taking this course.
n(A) = 13 n(A) means the number of students in this course that are male
n(B) = 30 n(B) means the number of students taking this course
InclusionExclusion Principle
this represent the number of students in this course that are male or are students in this course (includes males and females) UNION
this represent the number of students in this course that are male and are students in this course (includes males and females) INTERSECTION
Notice there students that are in both sets To avoid
counting these students twice We need to use the InclusionExclusion
Principle.
n(A) = 13  n(B) = 30 
Introduction of Venn Diagrams
Here is a very, very good link http://infinity.sequoias.cc.ca.us/faculty/woodbury/Stats/Tutorial/Sets_Venn2.htm
Here is a really cool java page http://www.stat.berkeley.edu/~stark/Teach/SticiGui/Lablets/Lablet7/lablet.htm
Here is a good example http://www.cs.uni.edu/~campbell/stat/prob2.html
When you are ready, click here for the assignment for this section.
Section 5.3 Venn Diagrams and Counting
Read the section. Use this link
Here is a very, very good link http://infinity.sequoias.cc.ca.us/faculty/woodbury/Stats/Tutorial/Sets_Venn2.htm
This is the type of problem I want you to understand.
A survey of 100 bank customers revealed that 58 of them have a savings account, 63 of them have a checking account, 22 of them have a savings account and a loan, 16 of them have a checking account and a loan, 27 of them have only a checking account, 12 have a savings account and a checking and a loan. Assume that every customer has a least one of the services.  
Lets organize our data. I tried color code it.  
Let U = {number of customers}  n(U) = 100  
Let C = { customers with checking accounts}  n(C) = 63  
Let S = { customers with savings accounts}  n(S) = 58  
Let L = { customers with loans}  n(L) = ? (don't know yet!)  

22  

16  

12  

27  
Lets try to answer 3 questions.  

Start with the easy information. Customers with savings account and checking account and loan = 12 (below)  Customers with a checking account only = 27 (below) 
Customers with checking account 
We are not given the number of customers with checking and saving
accounts. We know 63 customers have checking accounts. 63  (27 + 12 + 4) = 63  43 = 20 
Customers with savings account 
We know 58 customers have savings accounts. 58  (20 + 12 + 10) = 58  42 = 16 
We are not given the number of customers with loans We
know there are 100 customers. 
Looking at the Venn Diagram, can you answer these questions now?

When you are ready, click here for the assignment for this section.
Please notify me of any errors on this page joe@joemath.com
09/23/10