CS322

1. Week 5 - Friday CS322

2. Last time • What did we talk about last time? • Sequences • Summation and production notation • Proof by induction

3. Questions?

4. Logical warmup • Consider six straight silver chains made up of five links each • What if you want to make one large circular chain? • The jeweler will charge $1 for every link that he must cut open and then weld close • What is the cheapest price possible to make the six chains into one chain?

5. Review of Sequences

6. Explicit formulas • It is often possible to use a formula to describe the relationship between the value of a subscript and the value of the corresponding term • Example: ai = i2, for integers i 1

7. Summation notation • Summation notation is used to describe a summation of some part of a series • Expanded form is the summation written without the sigma

8. Product notation • Product notation is used to describe a product of some part of a series • Expanded form is the product written without the pi

9. Return to Mathematical Induction

10. Proof by mathematical induction • To prove a statement of the following form: • n  Z, where n a, property P(n) is true • Use the following steps: • Basis Step: Show that the property is true for P(a) • Induction Step: • Suppose that the property is true for some n = k, where k  Z, k a • Now, show that, with that assumption, the property is also true for k + 1

11. Example • Prove that, for all integers n  8, n = 3a + 5b, where a,b are integers greater than or equal to zero • Hint: Use induction and cases • This is the example given in the book • Another way to write it is that any amount of change above 8 cents can be made using only 3 cent and 5 cent coins

12. Geometric series • Prove that, for all real numbers r  1 • Hint: Use induction • This is the sum of a geometric sequence, also known as a geometric series • This result generalizes our example with the sum of powers of 2 • There is a small issue for r = 0, but Epp ignores it

13. More Examples

14. Divisibility • Prove that, for all integers n ≥ 1, 22n– 1 is divisible by 3 • Hint: Use induction

15. Inequality • Prove that, for all real number n ≥ 3, 2n + 1 < 2n • Hint: Use induction

16. Fibonacci • The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … • We can define it (recursively) as follows: • F1 = F2 = 1 • Fn = Fn-1 + Fn-2, n≥ 2 • Prove that, for all integers n ≥ 1, F4n is divisible by 3 • Hint: Use induction

17. Student Lecture Strong Induction

18. Strong Induction

19. Strong induction • There are situations where we need more than the fact that the kth element maintains some property to prove the k + 1st element has the same property • Strong induction allows us to use the kth element, the k-1st, element, the k-2nd element, and so on • This is usually most helpful when the subterms you are doing induction on are of unknown size

20. Proof by strong induction • To prove a statement of the following form: • n  Z, where n a, property P(n) is true • Use the following steps: • Basis Step: Show that the property is true for P(a), P(a+1), … P(b-1) , P(b), where a ≤ b, bZ • Induction Step: • Suppose that the property is true for some a ≤ i < k, where k  Z and k > b • Now, show that, with that assumption, the property is also true for k

21. Example Theorem: For all integers n  2, n is divisible by a prime Proof: • Basis step: (n = 2) The property is true for n = 2 because 2 is divisible by 2 • Induction step: Assume that all numbers i where 2 ≤ i < k are divisible by a prime, where k > 2 and k  Z • Case 1:k is prime • If k is prime, k = k∙1, therefore k is divisible by a prime, namely itself • Case 2: k is composite • If k is composite, k = a∙b, where a,b  Z and 2 ≤ a < k and 2 ≤ b < k • By the induction hypothesis, a is divisible by some prime p • Thus, k = p∙c∙b = p∙d and k is divisible by prime p • Since we have shown the basis step and induction step of strong mathematical induction, the claim is true QED

22. Example Theorem: It takes exactly n-1 steps to assemble a jigsaw puzzle with n pieces Proof: • Basis step: (n = 1) A puzzle with 1 piece takes 0 steps to put together • Induction step: Assume it takes i – 1 steps to put together puzzles of size i where 1 ≤ i < k, where k > 1 and k  Z • The last step in a puzzle of size k is putting together a subpuzzle of size j and a subpuzzle of size k – j where j  Z and 1 ≤ j < k and 1 ≤ k - j < k • By the induction hypothesis, it took j – 1 steps to put together one subpuzzle and k – j – 1 steps to put together the other • Thus, the total number of steps is j – 1 + k + j – 1 + 1 = k – 1 steps • Since we have shown the basis step and induction step of strong mathematical induction, the claim is true QED

23. Well-Ordering Principle for Integers • It turns out that the concept of truth through mathematical induction is equivalent to another principle • Well-Ordering Principle for the Integers: • Let S be a set containing one or more integers all of which are greater than some fixed integer. Then S has a least element.

24. Quiz

25. Upcoming

26. Next time… • Recursion • Second-order linear homogeneous recurrence relations with constant coefficients

27. Reminders • Homework 3 is due on Monday • Keep reading Chapter 5