MAT 2720Discrete Mathematics

Section 6.8

The Pigeonhole Principle

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Goals

The Pigeonhole Principle (PHP)•First Form

•Second Form

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

Example 1

Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.

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

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).

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:

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

Example 2

If 20 processors are interconnected, show that at least 2 processors are directly connected to the same number of processors.

MAT 2720Discrete Mathematics

Section 7.2

Solving Recurrence Relations

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Goals

Recurrence Relations (RR)•Definitions and Examples

•Second Order Linear Homogeneous RR with constant coefficients

Classwork

*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

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

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

Recurrence Relations (RR)

0 1 2 1

Given a sequence

, , , ,

is called a RR

n

n n

a

a f a a a a

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?

Visa Card Commercial Illustrations

Example 1: Population Model (1202)

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}.

Example 2(b)

A person invests $ 1000 at 12 percent interest compounded annually.

Find an explicit formula for An.

Example 2(c)*

RR is closed related to recursions / recursive algorithms

Example 2(c)*

RR is closed related to recursions / recursive algorithms

Recursions are like mentally ill people….

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?

Definitions

Second Order Linear Homogeneous RR with constant coefficients

1 1 2 2n n na c a c a

Example 3

1 2 0 15 6 ; 7, 16n n na a a a a Solve

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

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

Expectations

You are required to clearly show how the system of equations are being solved.

Verifications

How do I check that my formula is (probably) correct?

Generalized Method

The above method can be generalized to more situations and by-pass some of the steps.

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

Example 4

1 2 0 14 4 ; 1, 4n n na a a a a Solve

*The Theorem looks familiar?

Where have you seem a similar theorem?