1. MAT 2720 Discrete Mathematics
Section 2.5
Strong Induction

2. Recall: Principle of Mathematical Induction (PMI)
PMI: It suffices to show 1. P(1) is true. (Basic Step) 2. If P(k) is true, then P(k+1) is also true, for all k (Inductive Step)

3. Goals Principle of Strong Induction (PSI)
Why? When P(k) alone is not sufficient to prove P(k+1).
How? Use P(k-1), P(k-2),…etc.

4. Principle of Strong Induction (PSI)
PSI: It suffices to show
1. P(?) is/are true*.
2. If P(1), P(2),…, P(k) are true,
then P(k+1) is also true, for all k
*Adaptations for the Basic Step are required for different situations.

5. Adaptations – Example (a)
PSI: It suffices to show 1. P(?) is/are true How many initial cases?

6. Adaptations – Example (b)
PSI: It suffices to show 1. P(?) is/are true How many initial cases?

7. Adaptations – Example (c)
PSI: It suffices to show 1. P(?) is/are true How many initial cases?

8. Move to Page 2...
To save paper, I squeezed in something at the end of page 1.

9. Example 1
Prove that each integer greater than 1 is either prime or is a product of primes.

10. Before our next example,...
We need to make sure everyone have seen the ceiling and floor functions.
Typically, you have learned these functions in pre-calculus.
They have a lot of applications, especially in CS.
Go back to page 1, please.

11. Floor and Ceiling Functions (3.1)
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12. Floor and Ceiling Functions (3.1)
Visual Image

13. Floor and Ceiling Functions (3.1)
Visual Image

14. Lemma (HW)

15. Example 2
Practice…

16. Example 3 (Adaptation of the Basic Step)
Fibonacci Sequence is defined by

17. Group Explorations Some steps are provided.
Do the Inductive Step first.
Decide how many cases you need to prove in the Basic Step.

18. Group Explorations
Fibonacci Sequence is defined by