1. 1 CMSC 250 Chapter 4, con't., Inductive Proofs

2. 2 CMSC 250 Description l Inductive proofs must have: –Base case: where you prove that what it is you are trying to prove is true about the base case –Inductive hypothesis: where you state what will be assumed in the proof –Inductive step: show: –where you state what will be proven below proof: –where you prove what is stated in the show portion –this proof must use the inductive hypothesis somewhere

3. 3 CMSC 250 Example l Prove this statement: Base case ( n = 1): l Inductive hypothesis (assume the statement is true for n = p ): Inductive step (show the statement is true for n = p + 1), i.e, show :

4. 4 CMSC 250 Variations l 2 + 4 + 6 + 8 + … + 20 = ? l If you can, use the fact just proved, that: l Can it be rearranged into a form that works? l If not, it must be proved from scratch

5. 5 CMSC 250 Another example

6. 6 CMSC 250 Discrete Structures CMSC 250 Lecture 23 March 26, 2008

7. 7 CMSC 250 Another example- geometric progression

8. 8 CMSC 250 Another example- a divisibility property

9. 9 CMSC 250 A sequence example l Assume the following definition of a sequence: l Prove :

10. 10 CMSC 250 Discrete Structures CMSC 250 Lecture 24 March 28, 2008

11. 11 CMSC 250 An example with an inequality Prove this statement: Base case (n = 3): Inductive hypothesis (n = p): assume Inductive step (n = p + 1): Show:

12. 12 CMSC 250 Another example with an inequality

13. 13 CMSC 250 A less-mathematical example l If all we had was 2-cent coins and 5-cent coins, we could form any value greater than 3 cents. –Base case (n = 4): –Inductive hypothesis (n = p): –Inductive step (n = p + 1):

14. 14 CMSC 250 Discrete Structures CMSC 250 Lecture 25 March 31, 2008

15. 15 CMSC 250 Recurrence relation example l Assume the following definition of a function: l Prove the following definition property:

16. 16 CMSC 250 Strong induction l Regular induction: P(n)  P(n+1) l With strong induction, the implication changes slightly: –if the statement to be proven is true for all preceding elements, then it's true for the current element (  n)[(  i, a  i  n)[P(i)]  P(n+1)] l The strong induction principle: P(0)  …  P(p) (  n)[P(0)  P(1)  P(2)  …  P(n)  P(n + 1)]  (  n ≥ 0)[P(n)]

17. 17 CMSC 250 Now prove the recurrence relation property, using strong induction l Here's the function definition again: l This is the property to be proven:

18. 18 CMSC 250 Another recurrence relation example l Assume the following definition of a function: l Prove the following definition property, using strong induction:

19. 19 CMSC 250 Discrete Structures CMSC 250 Lecture 26 April 2, 2008

20. 20 CMSC 250 Another example- a divisibility property l Assume the following definition of a recurrence relation: l Prove using strong induction that all elements in this relation have this property:

21. 21 CMSC 250 Another example l Assume the following definition of a recurrence relation: l Prove using strong induction that all elements in this relation have this property:

22. 22 CMSC 250 Discrete Structures CMSC 250 Lecture 27 April 4, 2008

23. 23 CMSC 250 Another example Theorem: for all n ≥ 2 there exist primes p 1, p 2,…, p k, and exponents e 1, e 2,… e k, such that

24. 24 CMSC 250 Constructive induction l Show: (i.e., find integers A and B for which this is true) In particular, we want to find the smallest A and B which will work

25. 25 CMSC 250 A factorial example