Part 1
Part 2
Part 3
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Part 1
Recurrence Relations
Conditional Probability
Generalized Inclusion/ Exclusion
P(knowing 'union' problem)
P(knowing)
an+1 = 2an
rn+1 = 2rn
rn = r n
r=2
Part
SYLLABUS AND COURSE INFORMATION
SUUMMER 2014 SEMESTER
MAT 226 DISCRETE MATHEMATICS
INSTRUCTOR INFORMATION
Instructor: Dr. Terry Crites
FAX: 523-5847
Office: AMB 124
Office Phone: 523-6883
E-Mail: Terr
Permutations and Combinations
Permutations and Combinations
There is a group of seven people, including April, May, and June:
In how many different ways can you line up the seven people in a
row?
How
Counting Principles
Multiplication Principle
How possible ID numbers are there if an ID
consists of the numbers 1, 5 and 8?
How many possible ID numbers are there if
an ID consists of two letters foll
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Generalized (PHP) pigeon hole principle
If you have N objects in K boxes then there is some box with r = [N/K] objects
Note: "x" is the ceiling function also called the least integer f
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b) How many socks must be selected to guarantee 4 of the same color
2(4-1) + 1 = 7
c) How many socks must be drawn to guarantee 4 black socks
Not PHP 2(4-1) + 1 = 16?
6.3 Permutations
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Definition: A one-to-one function has for each element in the range only one
mapping from the domain.
Example
See picture
Example
How many functions f: A -> B are possible?
n * n n = n
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6.2 Pigeon-hole principle (PHP)
- h boxes
- k+1 objects
Some boxes must have at least two objects.
Example
6 objects 5 boxes
See picture
Example Birthdays
12 months 13 people
Example
H
Example NAU phones start with 523 or 522 and not 911
10,000,000 - 5
9,999,995
|P(S)| = 2|S|
Proof:
Let S = cfw_S1, S2, , Sn
|S| = n
Choices for subsets are
2 * 2 * * 2 = 2n = 2|S|
Example
1.
How many
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Definition: The universal set U is the set of all elements under consideration.
Example U = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10
A = cfw_1, 2, 3
B = cfw_ 3, 4, 5
Definition: The compleme
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Section 6.1 Counting
Sum Rule: if task 1 can be performed in 'n1' ways and task 2 in 'n2' ways, then
there are 'n1 + n2' ways for the procedure.
Example
4 store brands
2 name brands
Ho
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Mat 226
First Time Discrete Math
Sec 2.1/ 2.2 sets
Definition: A set is a collection of objects called elements
Notations/ definitions
Use capital letters for sets
'Epsilon' is an el
Proofs
Application of Pigeon Hole Principle
How many cards must be selected from a
standard deck of 52 cards to guarantee
that at least three cards of the same suit
are chosen?
Binomial Theorem
Prove