1. The Pigeonhole Principle 6/11/20151

2. The Pigeonhole Principle In words: –If n pigeons are in fewer than n pigeonholes, some pigeonhole must contain at least two pigeons 6/11/2015 2 What is n? http://www.blog.republicofmath.com/archives/3115

3. The Pigeonhole Principle In math: 6/11/2015 3

4. The Pigeonhole Principle What is a set? a finite set? What is |A|? What is a function? the domain of a function? the codomain of a function? Why say “distinct”? 6/11/2015 4

5. Applications of The Pigeonhole Principle In any group of 8 people, two were born on the same day of the week What are the “pigeons” and what are the “pigeonholes”? A = the set of people, B = {Sun, … Sat}, f(a) = the day of the week on which a was born 6/11/2015 5

6. Applications of The Pigeonhole Principle Suppose each pigeonhole contains one bird, and every bird moves to an adjacent square (up, down, left or right). Show that no matter how this is done, some pigeonhole winds up with at least 2 birds. 6/11/2015 6 DDDDD DDDDD DDDDD DDDDD DDDDD

7. Applications of The Pigeonhole Principle Suppose each pigeonhole contains one bird, and every bird moves to an adjacent square (up, down, left or right). Show that no matter how this is done, some pigeonhole winds up with at least 2 birds. 6/11/2015 7 DDDDD DDDDD DDDDD DDDDD DDDDD

8. Applications of The Pigeonhole Principle Suppose each pigeonhole contains one bird, and every bird moves to an adjacent square (up, down, left or right). Show that no matter how this is done, some pigeonhole winds up with at least 2 birds. 6/11/2015 8 DDDDD DDDDD DDDDD DDDDD DDDDD