The Pigeonhole Principle 03/22/19 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 03/22/19 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 ) 03/22/19 The Pigeonhole Principle • • • • • • • 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”? 03/22/19 Applications of The Pigeonhole Principle • In any group of 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 03/22/19 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 birds 03/22/19 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 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 birds 03/22/19 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 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 birds φ(α) = τηε σθυαρε α µ οϖεστο Α = 13, Β = 12 03/22/19 ... 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 ) 03/22/19 The Pigeonhole Principle. ..The Pigeonhole Principle • In words: – If n pigeons are in fewer than n pigeonholes, some pigeonhole must contain... function? the codomain of a function? Why say “distinct”? 03/22/19 Applications of The Pigeonhole Principle • In any group of people, two were born on the same day of the week • What are the “pigeons”