Finite Mathematics Lesson 5
Sections 5.1 - 5.3

 Section 5.1 Section 5.2 Section 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 set collection of A = {1,2,b}  set A contains 3 elements subset all elements are contained in another set A = {1, 2, b}  B = {1, b }  B is a subset of A universal set set containing all elements U ={ all fifty states} complement 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} empty set set containing no elements  - sometimes called the null set     � �  is a subset of all sets intersection set containing elements that belong to both set A and set B, If A = {1, 3, 5} and B ={3, 4, 6 } , then = {3} union 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} 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

Inclusion-Exclusion 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

Inclusion-Exclusion 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 Inclusion-Exclusion 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 Section 5.3   Venn Diagrams and Counting

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!) saving account and loan 22 checking account and loan 16 savings account and checking account and loan 12 checking account only (i.e.  not loan and not savings) 27 Lets try to answer 3 questions. What is the number of customers  who have a checking or savings account but no loans?  What is the number of customers  who have a checking but no savings account ?   What is the number of customers who have a loan?
 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 and a loan 16 - 12  = 4 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 and a loan 22 - 12 = 10 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. The number of customers who only have loans is  100  - (27+20+16+4+12+10) = 100 - 89 = 11 Looking at the Venn Diagram,  can you answer these questions now?

 What is the number of customers  who have a checking or savings account but no loans? 27 + 20 + 16 =  63 What is the number of customers  who have a checking but no savings account ? 27 + 4 =  31 What is the number of customers who have a loan?  4 + 12 +10 + 11 = 37 