1. COMP 170 L2 Page 1

2. COMP 170 L2 Page 2

3. COMP 170 L2 L10: Intro to Induction l Objective n Introduce induction from proof-by-smallest-counter-example  Making use of small-problem/big-problem relationships in proofs Page 3

4. COMP 170 L2 Recap: 2010-03-30 l Given: (a) p(0) is True; (b) If n>0, then p(n-1) => p(n) l Conclusion: p(n) is True for all l Proof by Smallest Counter Example Page 4

5. COMP 170 L2 Outline l Weak induction l Strong induction Page 5

6. COMP 170 L2 Weak Principle of Math Induction l We have actually proved: Page 6

7. COMP 170 L2 Weak Principle of Math Induction l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(1) => p(2) n p(2) => p(3), n …. n So, p(n) is True for all n>=0 Page 7

8. COMP 170 L2 Inductive Proof l Base Case (n=b): Show that p(b) is True l Induction (n>b): Show p(n-1) => p(n) n Induction Hypothesis  p(n-1) is True n Inductive Step:  p(n) is True l Inductive conclusion: n p(n) is True for all n>=b Page 8

9. COMP 170 L2 Page 9

10. COMP 170 L2 Page 10

11. COMP 170 L2

12. Outline l Weak induction l Strong induction Page 12

13. COMP 170 L2 Review: Weak Induction l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(1) => p(2) n p(2) => p(3), n …. n So, p(n) is True for all n>=0 Page 13

14. COMP 170 L2 l Suppose b=0 l Intuitively n p(0) n p(0) => p(1) n p(0) /\ p(1) => p(2) (don’t have p(1) => p(2)) n p(0) /\ p(1) /\ p(2) => p(3), (don’t have p(2) => p(3)) n …. n So, p(n) is True for all n>=0 Page 14

15. COMP 170 L2 Page 15

16. COMP 170 L2 Strong Induction Implicitly Used in Proof of Euclid’s Division Theorem

17. COMP 170 L2 Remarks Page 17

18. COMP 170 L2 Summary Page 18