1. Miniconference on the Mathematics of Computation
MTH 210
Strong Induction
Dr. Anthony Bonato
Ryerson University

2. Example Eg: given a0 = 0, a1 = 4 and ak = 6ak-1-5ak-2
want to show an = 5n-1 for all n ≥ 0

3. Strong Induction
same as induction, but assume P(n) is true for ALL values up to a given k.
don’t only assume true for k-1
strong induction is really the same as induction, just the induction hypothesis is stated differently

4. Example, continued eg Given a0 = 0, a1 = 4 and ak = 6ak-1 - 5ak-2.
Want to show an = 5n - 1.
NOTE: ak depends on two previous values, not just one

5. Exercises

6. Miniconference on the Mathematics of Computation
MTH 210
Counting
Dr. Anthony Bonato
Ryerson University

7. Independent events
two events are independent if they do not interact with each other eg
2-digit numbers
first digit is an event, second digit is an event
choosing a first digit won’t effect the choice of the second

8. Counting independent events
suppose you have k independent events
n1 objects from Event 1
n2 objects from Event 2
n3 objects from Event 3
… nk objects from Event k
then the number of objects in every event
Is: n1n2 …nk

9. Example, continued 10 possibilities for first digit
10 possibilities for second digits
digits are independently chosen
so there are 10 x 10 = digit numbers
NB: allow “02”, “05”, etc.

10. Pigeonhole principle
Idea: more pigeons than holes, then at least one hole has two or more pigeons

11. Pigeonhole property More formal statement:
If you have n+1 objects assigned to n properties, then at least two objects have the same properties

12. Exercises