4/9/2020
1

The Pigeonhole Principle
• In words:
– If n pigeons are in fewer than n pigeonholes, some pigeonhole must contain at least two pigeons n
What is n ?
http://www.blog.republicofmath.com/archives/3115
2
4/9/2020

The Pigeonhole Principle
• In math:
Let f : A

B , where A and B are finite sets and A

B .
Then there exist distinct elements a
1
, a
2

A such that f ( a
1
)
 f ( a
2
).
4/9/2020
3

The Pigeonhole Principle
• What is a set?
• a finite set?
• What is |A|?
• What is a function?
Let f : A

B , where A and B are finite sets and A

B .
Then there exist distinct elements a
1
, a
2

A such that f ( a
1
)
 f ( a
2
).
• the domain of a function?
• the codomain of a function?
• Why say “distinct”?
4
4/9/2020

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
5
4/9/2020

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.
D D D D D
D D D D D
D D D D D
D D D D D
D D D D D
6
4/9/2020

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.
D D D D D
D D D D D
D D D D D
D D D D D
D D D D D
7
4/9/2020

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.
D D D D D
D D D D D
D D D D D
D D D D D
D D D D D
A

birds on red squares
B

gray squares f ( a )

the square a moves to
A

13, B

12
8
4/9/2020