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}

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. 

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

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!)
  • 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.
  1. What is the number of customers  who have a checking or savings account but no loans? 
  2. What is the number of customers  who have a checking but no savings account ?  
  3. 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)
venn1.gif (2746 bytes) venn2.gif (2830 bytes)

 

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
venn3.gif (3468 bytes) venn4.gif (3552 bytes)

 

Customers with savings account
and a loan
22 - 12 = 10

We know 58 customers have savings accounts.
58  -   (20 + 12 + 10) = 58 - 42 = 16
venn5.gif (3624 bytes) venn7.gif (3708 bytes)

 

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

venn6.gif (3740 bytes)

Looking at the Venn Diagram,  can you answer these questions now?

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

When you are ready, click here for the assignment for this section.

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Please notify me of any errors on this page   joe@joemath.com

09/23/10