MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .
-
Upload
amari-trimm -
Category
Documents
-
view
229 -
download
7
Transcript of MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .
![Page 1: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/1.jpg)
MAT 2720Discrete Mathematics
Section 6.8
The Pigeonhole Principle
http://myhome.spu.edu/lauw
![Page 2: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/2.jpg)
Goals
The Pigeonhole Principle (PHP)•First Form
•Second Form
![Page 3: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/3.jpg)
The Pigeonhole Principle (First Form)
If n pigeons fly into k pigeonholes and k<n, some pigeonhole contains at least two pigeons.
1st
3x1xnx2x
2nd 3rd k- th
4x
![Page 4: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/4.jpg)
Example 1
Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.
![Page 5: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/5.jpg)
Example 1
Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.
Spades Hearts Diamonds Clubs
4C 5C
1C2C
3C
![Page 6: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/6.jpg)
Example 1
Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.We can think of the 5 cards as 5 pigeons and the 4 suits as 4 pigeonholes. By the PHP, some suit ( pigeonhole) is assigned to at least two cards ( pigeons).
![Page 7: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/7.jpg)
Example 1
Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.
Formal Solutions:
![Page 8: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/8.jpg)
The Pigeonhole Principle (Second Form)
1 2 1 2 1 2
If : and ,
then , such that and ( ) ( ).
f X Y X Y
x x X x x f x f x
X Y
1x
2x
![Page 9: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/9.jpg)
Example 2
If 20 processors are interconnected, show that at least 2 processors are directly connected to the same number of processors.
![Page 10: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/10.jpg)
MAT 2720Discrete Mathematics
Section 7.2
Solving Recurrence Relations
http://myhome.spu.edu/lauw
![Page 11: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/11.jpg)
Goals
Recurrence Relations (RR)•Definitions and Examples
•Second Order Linear Homogeneous RR with constant coefficients
Classwork
![Page 12: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/12.jpg)
*Additional Materials…
We will cover some additional materials that may not make senses to all of you.
They are for educational purposes only, i.e. will not appear in the HW/Exam
![Page 13: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/13.jpg)
2.5 Example 3
0 1
1 2
0, 1
2n n n
f f
f f f n
Fibonacci Sequence is defined by
Show that
2 , nnf n
![Page 14: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/14.jpg)
2.5 Example 3
0 1
1 2
0, 1
2n n n
f f
f f f n
Fibonacci Sequence is an example of RR.
RRI nitial Conditions
![Page 15: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/15.jpg)
Recurrence Relations (RR)
0 1 2 1
Given a sequence
, , , ,
is called a RR
n
n n
a
a f a a a a
![Page 16: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/16.jpg)
Example 1: Population Model (1202) Suppose a newly-born pair of rabbits, one
male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits.
Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on.
How many pairs will there be in one year?
![Page 17: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/17.jpg)
Visa Card Commercial Illustrations
![Page 18: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/18.jpg)
Example 1: Population Model (1202)
![Page 19: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/19.jpg)
Example 2(a)
A person invests $ 1000 at 12 percent interest compounded annually.
If An represents the amount at the end of n years, find a recurrence relation and initial conditions that define the sequence {An}.
![Page 20: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/20.jpg)
Example 2(b)
A person invests $ 1000 at 12 percent interest compounded annually.
Find an explicit formula for An.
![Page 21: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/21.jpg)
Example 2(c)*
RR is closed related to recursions / recursive algorithms
![Page 22: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/22.jpg)
Example 2(c)*
RR is closed related to recursions / recursive algorithms
Recursions are like mentally ill people….
![Page 23: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/23.jpg)
Example 1
0 1
1 2
0, 1
2n n n
f f
f f f n
Fibonacci Sequence
How to find an explicit formula?
![Page 24: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/24.jpg)
Definitions
Second Order Linear Homogeneous RR with constant coefficients
1 1 2 2n n na c a c a
![Page 25: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/25.jpg)
Example 3
1 2 0 15 6 ; 7, 16n n na a a a a Solve
![Page 26: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/26.jpg)
Recall Example 2
A person invests $ 1000 at 12 percent interest compounded annually.
1
0
1.12
1.12
n n
n
n
A A
A A
ntI n the f orm of Depends on I nitial Conditions
![Page 27: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/27.jpg)
Example 3
From last the example, it makes sense to attempt to look for solutions of the form
Where t is a constant.
1 2 0 15 6 ; 7, 16n n na a a a a
nna kt
Solve
![Page 28: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/28.jpg)
Expectations
You are required to clearly show how the system of equations are being solved.
![Page 29: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/29.jpg)
Verifications
How do I check that my formula is (probably) correct?
![Page 30: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/30.jpg)
Generalized Method
The above method can be generalized to more situations and by-pass some of the steps.
![Page 31: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/31.jpg)
Theorem
Second Order Linear Homogeneous RR with constant coefficients
Characteristic Equation
1. Distinct real roots t1,t2 :
2. Repeated root t :
1 1 2 2n n na c a c a
21 2t c t c
1 2n n
na b t d t
n nna b t d n t
![Page 32: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/32.jpg)
Example 4
1 2 0 14 4 ; 1, 4n n na a a a a Solve
![Page 33: MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle .](https://reader036.fdocuments.us/reader036/viewer/2022062223/5519be375503467a578b4a95/html5/thumbnails/33.jpg)
*The Theorem looks familiar?
Where have you seem a similar theorem?