07-Pigeonhole.pdf - Pigeonhole Principal and Complexity Theory Cassie Kierzkowski Monday 10th September 2018 Sections covered 4.1 4.2 4.3 4.4 Midterm 1

07-Pigeonhole.pdf - Pigeonhole Principal and Complexity...

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Pigeonhole Principal and Complexity Theory Cassie Kierzkowski Monday 10 th September, 2018 Sections covered: 4.1, 4.2, 4.3, 4.4 Midterm 1: Thursday September 13th. Will cover: HW1 through HW3, which is chapters 2, 3, and section 4.1. 1 Pigeonhole Principle 1.1 Strong Formulation Theorem 1. Let X and Y be sets with | Y | = m and | X | = n . Let q = d n m e . Let f be a function from X to Y . If r > 0 then there must exist an element y Y and distinct elements x 1 , ..., x q such that f ( x 1 ) = ... = f ( x q ) = y . Example: There are 50 baskets of apples, each basket contains no more than 24 apples. Show that there exists at least 3 baskets with the same number of apples. Example: Show that for any non-negative integer n there exists a number, x , composed of only the digits 5 and 0 such that n divides x . 1
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Cassie Kierzkowski Pigeonhole Principal and Complexity Theory Example: Say you have 56 squares of side length 1 inside a circle of radius 2. Show that at least 5 squares must have a common point. (For sake of a
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