01Pigeonhole principle discrrete mathematics for computer science

The Pigeonhole Principle... 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?. Appl

The Pigeonhole Principle

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 σετσ ανδ Α > Β

Τηεν τηερε εξιστ διστινχτ ελεµ εντσ

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

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”?

Let f : A → Β , ωηερε Α ανδ Β αρε φινιτε σετσ ανδ Α > Β Τηεν τηερε εξιστ διστινχτ ελεµ εντσ

α1, α2 ∈ Α συχη τηατ φ (α1) = φ (α2).

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

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

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

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