MAT 2720 Discrete Mathematics Section 6.8 The Pigeonhole Principle http://myhome.spu.edu/lauw

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.

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.

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)

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

MAT 2720 Discrete Mathematics Section 7.2 Solving Recurrence Relations http://myhome.spu.edu/lauw

Goals Recurrence Relations (RR) Classwork 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 Fibonacci Sequence is defined by

2.5 Example 3 Fibonacci Sequence is an example of RR.

Recurrence Relations (RR)

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?

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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 Fibonacci Sequence How to find an explicit formula?

Definitions Second Order Linear Homogeneous RR with constant coefficients

Example 3 Solve

Recall Example 2 A person invests $ 1000 at 12 percent interest compounded annually.

Example 3 Solve From last the example, it makes sense to attempt to look for solutions of the form Where t is a constant.

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 :

Example 4 Solve

*The Theorem looks familiar? Where have you seem a similar theorem?