MAT 2720 Discrete Mathematics

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

2. Goals • The Pigeonhole Principle (PHP) • First Form • Second Form

3. The Pigeonhole Principle (First Form) If n pigeons fly into k pigeonholes and k<n, some pigeonhole contains at least two pigeons.

4. Example 1 Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.

5. Example 1 Prove that if five cards are chosen from an ordinary 52- card deck, at least two cards are of the same suit.

6. 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).

7. 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:

8. The Pigeonhole Principle (Second Form)

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

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

11. Goals • Recurrence Relations (RR) • Definitions and Examples • Second Order Linear Homogeneous RR with constant coefficients • Classwork

12. *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

13. 2.5 Example 3 Fibonacci Sequence is defined by

14. 2.5 Example 3 Fibonacci Sequence is an example of RR.

15. Recurrence Relations (RR)

16. 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?

17. Visa Card Commercial Illustrations

18. Example 1: Population Model (1202)

19. 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}.

20. Example 2(b) • A person invests $ 1000 at 12 percent interest compounded annually. • Find an explicit formula for An.

21. Example 2(c)* • RR is closed related to recursions / recursive algorithms

22. Example 2(c)* • RR is closed related to recursions / recursive algorithms • Recursions are like mentally ill people….

23. Example 1 Fibonacci Sequence How to find an explicit formula?

24. Definitions Second Order Linear Homogeneous RR with constant coefficients

25. Example 3 Solve

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

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

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

29. Verifications • How do I check that my formula is (probably) correct?

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

31. Theorem Second Order Linear Homogeneous RR with constant coefficients Characteristic Equation 1. Distinct real roots t1,t2 : 2. Repeated root t:

32. Example 4 Solve

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