Discrete Structures Chapter 5 Pigeonhole Principle Nurul Amelina Nasharuddin Multimedia Department

2       If n pigeons fly into m pigeonholes and n > m, then at least one hole must contain two or more pigeons Basic Form of the Pigeonhole Principle: A function from one finite set to a smaller finite set cannot be one-to-one: there must be at least two elements in the domain that have the same image in the co-domain Pigeonhole Principle

3 An arrow diagram for a function from a finite set to a smaller finite set must have at least two arrows from the domain that point to the same element in the codomain. Next is some of the examples on how to apply the pigeonhole principle.

4 In a group of 6 people, must there be at least 2 who were born in the same month? Not. Six people could have birthdays in each of the six months Jan to June Example (1)

5 In a group of thirteen people, must there be at least two who were born in the same month? Yes. For there are only 12 months in a year and number of people  number of months (13  12) Define a function B from the set of people to the set of 12 months B(x i ) = birth month of x i. Example (2)

6 A drawer contains 10 black and 10 white socks. You pull some out without looking at them. What is the least number of socks you must pull out to be sure to get a matched pair? Three. If you just pick two, they may have different colors. Define a function C that maps each sock to its color, i.e. C(s i ) = color of s i Example (3)

7 Let A = {1,2,3,4,5,6,7,8} (a) If five integers are selected from A, must at least one pair of the integers have a sum of 9? Yes. Partition the set A into {1,8}, {2,7}, {3,6}, and {4,5}. The function P is defined by letting P(a i ) be the subset that contains a i. (b) If four are selected? No. Example (4)

8 If n pigeons fly into m pigeonholes and for some positive integer, k, n  km, then at least one pigeonhole contains k + 1 or more pigeons. Generalized Form of Pigeonhole Principle: For any function f from a finite set X to a finite set Y and for any positive integer k, if n(X) > k  n(Y), then there is some y  Y such that y is the image of at least k+1 distinct elements of X. i.e. f(x 1 )=f(x 2 )= …=f(x k+1 )=y. Generalized Pigeonhole Principle

9 1.A hall has a seating capacity of 700. Must there be two people seated in the hall who have the same first and last initials? Why? 2.Let S = {3,4,5,6,7,8,9,10,11,12}. Suppose six integers are chosen from S. Must there be two integers whose sum is 15? Why? Quiz 4B Send your answers in the next class!