01Pigeonhole principle discrrete mathematics for computer science

The Pigeonhole Principle

03/22/19

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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?
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The Pigeonhole Principle
• In math:
Let f : A → Β, ωηερε Α ανδ Β
αρε φινιτε σετσανδ Α > Β .
Τηεν τηερε εξιστ διστινχτ ελεµ εντσ

α1 , α2 ∈ Α συχη τηατ φ(α1 ) = φ(α2 ).

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The Pigeonhole Principle
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Let f : A → Β, ωηερε Α ανδ Β

What is a set?
αρε φινιτε σετσανδ Α > Β .
a finite set?
Τηεν τηερε εξιστ διστινχτ ελεµ εντσ
α1 , α2 ∈ Α συχη τηατ φ(α1 ) = φ(α2 ).
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.

03/22/19

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Applications of The Pigeonhole
Principle
D D D D D
• Suppose each
pigeonhole contains
D D D D D
one bird, and every bird D D D D D
moves to an adjacent
D D D D D
square (up, down, left
or right). Show that no
D D D D D
matter how this is done, A = βιρδσον ρεδ σθυαρεσ
some pigeonhole winds Β = γραψ σθυαρεσ
up with at least 2 birds. φ(α) = τηε σθυαρε α µ οϖεστο
Α = 13, Β = 12
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