Transcript Pigeonhole

The Pigeonhole Principle
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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
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The Pigeonhole Principle
• In math:
Let f : A  B, where A and B
are finite sets and A  B .
Then there exist distinct elements
a1 ,a2 A such that f (a1 )  f (a2 ).
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The Pigeonhole Principle
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Let f : A  B, where A and B
What is a set?
are finite sets and A  B .
a finite set?
Then there exist distinct elements
a1 ,a2 A such that f (a1 )  f (a2 ).
What is |A|?
What is a function?
the domain of a function?
the codomain of a function?
Why say “distinct”?
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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
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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.
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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.
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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.
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A  birds on red squares
B  gray squares
f (a)  the square a moves to
A  13, B  12
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