MAT 2720 Discrete Mathematics

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MAT 2720Discrete Mathematics

• Section 6.8
• The Pigeonhole Principle

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Goals

• The Pigeonhole Principle (PHP)
• First Form
• Second Form

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The Pigeonhole Principle (First Form)

• If n pigeons fly into k pigeonholes and kltn, some
  pigeonhole contains at least two pigeons.

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Example 1

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

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Example 1

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

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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).

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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

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The Pigeonhole Principle (Second Form)
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Example 2

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

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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

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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

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2.5 Example 3

• Fibonacci Sequence is defined by

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2.5 Example 3

• Fibonacci Sequence is an example of RR.

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Recurrence Relations (RR)
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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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Visa Card Commercial Illustrations
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Example 1 Population Model (1202)
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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.

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Example 2(b)

• A person invests 1000 at 12 percent interest
  compounded annually.
• Find an explicit formula for An.

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Example 2(c)

• RR is closed related to recursions / recursive
  algorithms

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Example 2(c)

• RR is closed related to recursions / recursive
  algorithms
• Recursions are like mentally ill people.

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Example 1

• Fibonacci Sequence
• How to find an explicit formula?

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Definitions

• Second Order Linear Homogeneous RR with constant
  coefficients

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Example 3
Solve
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Recall Example 2

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

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Example 3
Solve

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

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Expectations

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

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Verifications

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

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Generalized Method

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

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Theorem

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

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Example 4

• Solve

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The Theorem looks familiar?

• Where have you seem a similar theorem?