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

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

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

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

Exercises

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

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

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

Example, continued 10 possibilities for first digit 10 possibilities for second digits digits are independently chosen so there are 10 x 10 = 100 2-digit numbers NB: allow “02”, “05”, etc.

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

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

Exercises