1. Chapter 4 4.1 Mathematical Induction 4.2 Strong Induction and Well-Ordering 4.3 Recursive Definitions and Structural Induction 4.4 Recursive Algorithms 4.5 Program Correctness 1

2. 4.1 Induction A very special rule of inference! Definition: A set S is well ordered if every subset has a least element. Note: [0, 1] is not well ordered since (0,1] does not have a least element. Examples: N is well ordered (under the ≤ relation) The set of finite strings over an alphabet using lexicographic ordering is well ordered. 2

3. Let P(x) be a predicate over a well ordered set S. The problem is to prove the rule of inference called The (first) principle of Mathematical induction In the case that S=N, the nature numbers, the principle has the following form. 3 4.1 Induction

4. The hypotheses are H1: P(0),and H2: P(n)→ P(n+1) for n arbitrary. H1 is called The Basis Step. H2 is called The Induction (Inductive) Step. We first prove that the predicate is true for the smallest element of the set S (0 if S=N) We then show if it is true for an element x (n if S=N) implies it is true for the “next” element in the set (n+1 if S=N). 4 4.1 Induction

5. Then Knowing it is true for the first element means it must be true for the element following the first or the second element Knowing it is true for the second element implies it is true for the third, and so forth. Therefore, induction is equivalent to modus ponens applied an countable number of times!! 5 4.1 Induction

6. FIGURE 1 Climbing and Infinite Ladder. 6

7. We can use the Principle to prove more general assertions because N is well ordered. Suppose we wish to prove for some specific integer k Now we merely change the basis step to P(k) and continue. Example: (a classic) To prove Example 10: Use mathematical induction to prove the following generalization of one of De Morgan’s Laws: whenever A 1, A 2,..,A n are subsets of a universal set U and n ≥ 2. 7 4.1 Induction

8. Example 13: Let n be a positive integer. Show that every 2 n X2 n checkerboard with one square removed can be tiled using right triominoes, where these pieces cover three squares at a time, as shown below. 8 4.1 Induction

9. 9