Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Notes: Chapter 2 Section 2.3: The Pigeonhole Principle Definition 1. If more than n · k objects are placed into n categories, then there exists a category with more than k objects in it. Example 1. A drawer with 8 black socks and 11 blue socks. It’s dark and you can’t tell them apart. How many must you take to guarantee 2 of the same color? 2 black? 2 blue? Example 2. A society of friends. Given a set S of people, there must be 2 people with the same number of friends. Example 3. There are 30 balls with the numbers 1 through 30 written on them. If 18 balls are drawn, there must be a pair whose sum is 35. Proof. The categories are the pairs that have a sum of is 35, plus the single balls that are not part of any pair, 1,2,3,4. ¤ Example 4. Let S ⊆ [3n], with |S| = 2n + 1. Then there will be 3 consecutive numbers. Proof. To avoid consecutive, must have one only in each of the categories: C1 = {1, 2, 3}, C2 = {4, 5, 6}, C3 = {7, 8, 9}, . . . , Cn = {3n − 2, 3n − 1, 3n}. But there are n categories and more than 2n objects, so one category must contain 3 objects. ¤ 1