The Pigeonhole Principle

Transcription

1 The Pigeonhole Principle Lecture 45 Section 9.4 Robb T. Koether Hampden-Sydney College Mon, Apr 16, 2014 Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

2 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

3 Outline 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

4 The Pigeonhole Principle The Pigeonhole Principle (Version 1) If n > m and you put n pigeons into m pigeonholes, then at least at least two pigeons are in the same pigeonhole. The Pigeonhole Principle (Version 2) If A and B are finite sets and A > B and f : A B, then f is not one-to-one. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

5 Example If a drawer contains 10 black socks and 10 blue socks, how many socks must you draw at random in order to guarantee that you have two socks of the same color? 20? 19? 11? 3? 2? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

6 Example A bag of jellybeans contains dozens of jelly beans of each of 8 different colors. How many jellybeans must we choose in order to guarantee that we have at least two jellybeans of the same color? To guarantee three of the same color? To guarantee four of the same color? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

7 Example If we choose 6 distinct integers from 1 to 9, At least one pair of them adds to 10. Why? At least two pairs of them have the same total. Why? How many integers must we choose from 1 to 99 in order to guarantee that at least two distinct pairs of them will have the same total? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

8 Outline 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

9 Functions on Finite Sets Theorem Let A and B be finite sets with A = B and let f : A B. Then f is one-to-one if and only if f is onto. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

10 Functions on Finite Sets Proof. Suppose that f is not onto. Then there exists y B such that y / f (A). So f (A) < B = A. However, restrict the codomain of f to f (A) and we have f : A f (A). By the Pigeonhole Principle, f is not one-to-one. Thus, if f is one-to-one, then f is onto. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

11 Functions on Finite Sets Proof. Suppose that f is not one-to-one. Then f (A) < A = B. Therefore, f (A) B. Thus, f is not onto. Therefore, if f is onto, then f is one-to-one. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

12 Decimal Expansions Theorem Let n m be a rational number in reduced form. If the decimal expansion of n m has not terminated after m significant digits, then it is a repeating decimal expansion. Furthermore, the repeating part of n m can have length no more than m. Find the decimal expansions of 1 16 and Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

13 Outline 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

14 The Generalized Pigeonhole Principle The Generalized Pigeonhole Principle Let A and B be sets and let f : A B. If A > m B for some integer m, then there exists an element y B that is the image of at least m + 1 elements of A. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

15 Example In a group of 85 people, at least 4 must have the same last initial. If we select 30 distinct numbers from 1 to 100, there must be at least three distinct pairs that have the same sum. What if we select 45 distinct numbers from 1 to 100? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

16 Outline 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

17 The Locker Door Problem Suppose we have a hallway with 1000 lockers, labeled 1 through Each locker door is closed. We also have 1000 students, labeled 1 through We send each student down the hallway with the following instructions: If you are Student k, then reverse the state of every k-th door, beginning with Door k. After all the students have been sent down the hallway, which doors will be open? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

18 The Locker Door Problem Suppose that we have only 10 doors and 10 students and that we wish to leave Doors 2, 4, 5, 8, and 9 open and the others closed. Which students should be send down the hallway? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

19 The Locker Door Problem Clearly, we must not send Student 1 down the hallway. Door 2 is closed, so we must send Student 2 down the hallway. Door 3 is closed, so we must not send Student 3 down the hallway. Door 4 is open, so we must not send Student 4 down the hallway. Door 5 is closed, so we must send Student 5 down the hallway. And so on. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

20 The Locker Door Problem Let D {1, 2, 3,..., 1000}. Can we choose a set S of students so that if we send them down the hallway, the doors in D will be open and the rest will be closed? Let S 1, S 2 be two distinct subsets of {1, 2, 3,..., 1000}. If we send the students in S 1 down the hallway, will a different set of doors be left open than if we had sent the students in S 2 down the hallway? Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

21 The Locker Door Problem Let A = {1, 2, 3,..., 1000}. Define f : P(A) P(A) as follows. For any set S A, let f (A) be the set of doors left open after the students in S have been sent down the hallway. Prove that f is a one-to-one correspondence. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

22 Outline 1 The Pigeonhole Principle 2 Functions on Finite Sets 3 The Generalized Pigeonhole Principle 4 The Locker Door Problem 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23

23 Assignment Assignment Read Sections 9.4, pages Exercises 2, 4, 8, 13, 18, 19, 21, 25, 27, 30, 34, 37, page 563. Robb T. Koether (Hampden-Sydney College) The Pigeonhole Principle Mon, Apr 16, / 23