MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Thursday, 4/17/14 The Addition Principle The Inclusion-Exclusion Principle The Pigeonhole Principle Reading: [J] 6.1, 6.8 [H] 3.5, 12.3 Exercises: [H] pp. 81-82: 1-6 [J] pp. 328-329: 1-8, p. 331: 28 Thursday, 4/17/14, Slide #1
Adding sizes of sets What is wrong with the following formula, where and are any two finite sets? = + Give an example where it s correct and another where it s incorrect. Thursday, 4/17/14, Slide #2
The Addition Principle For sets: If and are two sets, and =, then = + For events: If an event can occur in ways, and an event can occur in ways, and A and B cannot occur at the same time, then the number of ways that event A or event B can occur is: + If events A and B cannot occur at the same time, they are called mutually exclusive. Thursday, 4/17/14, Slide #3
The Inclusion-Exclusion Principle For sets: If and are two sets, then = + For events: If an event can occur in ways, and an event can occur in ways, and there are can occur at the same time, then the number of ways that event A or event B can occur is: + Thursday, 4/17/14, Slide #4
Example Suppose we roll a pair of 6-sided dice: How many possible rolls are there? How many ways are there for the dice to be equal ( doubles ) or to add up to 9? How many ways are there for the dice to be equal ( doubles ) or to add up to 10? Thursday, 4/17/14, Slide #5
Standard decks of cards 52 cards in 4 suits: 2 Black Suits: Spades ( ) and Clubs ( ) 2 Red Suites: Hearts ( ), Diamonds ( ) 13 cards in each suit: Ace > King > Queen > Jack > 10 > 9 > 8 > 7 > 6 > 5 > 4 > 3 > 2 King, Queen, & Jack are face cards Thursday, 4/17/14, Slide #6
Clicker Question From a standard deck of cards, how many ways are there to choose 2 cards so that either both are hearts or both cards are face cards. A. 25 B. 66 C. 78 Remember:, = ( )/ D. 141 E. 144 Thursday, 4/17/14, Slide #7
More Inclusion-Exclusion Examples How many numbers from 0 to 100 are divisible by 6 or by 8? How many possible zip codes are there that begin with 1 or end with 1? Can we generalize the inclusion-exclusion formula to 3 sets, A, B, C? I.e., How do we correct + + to avoid counting things more than once? Thursday, 4/17/14, Slide #8
Pigeonhole Principle Example Suppose we have people in a room, > 0, and some pairs of people shake hands. It s possible that anybody shakes hands with any number of other people, from 0 to 1 people. Prove: There must be two people who shake the same number of hands. Hint: Use two cases, and categorize each person by the number of hands she/he shakes. Case 1 is when everybody shakes at least one hand. Case 2 is when there is at least one person who shakes no hands. Thursday, 4/17/14, Slide #9
The Pigeonhole Principle phrased 3 ways Pigeonhole Principle (bird version): If pigeons each fly into one of h holes, and >h, then some hole must contain at least two pigeons. Pigeonhole Principle (shoebox version): If shoes are each placed into one of h boxes, and >h, then some box must contain at least two shoes. Pigeonhole Principle (category version): If objects are divided into h categories, and >h, then at least two objects must be in the same category. Thursday, 4/17/14, Slide #10
Examples In using the Pigeonhole Principle, it s important to be clear what the objects are, and what the categories/boxes are. In each example, first say what the objects and categories are. My mitten drawer has 10 pairs of mittens, all different. How many might I have to remove at random, before I m sure to get a matching pair? Let n be a positive integer, and suppose we have k > n arbitrary positive integers. Prove that there must be two of them whose difference is divisible by n. Hint: Think about the k numbers mod n. Let n be a positive integer. Prove that n must have some multiple each of whose decimal digits is either 0 or 1. Hint: Think about the last exercise with the numbers 1, 11, 111, 1111,. Thursday, 4/17/14, Slide #11
Strengthening the Pigeonhole Principle Example: Our class has 28 students. The Pigeonhole Principle says there must be at least two students who were born in the same month. But we can say more: There must be at least three born in the same month. Why? Suppose we had 50 students. What s the least number that are born in the same month. Thursday, 4/17/14, Slide #12
The Strong Pigeonhole Principle Strong Pigeonhole Principle: If p objects are placed into h boxes, then some box contains at least objects. Proof by contradiction: Suppose every box has < objects. Add up all the objects in all the boxes. Note: If an integer, then, so we can replace with in the theorem. Thursday, 4/17/14, Slide #13
Clicker Question Suppose 200 people exit a theater and walk to their cars, but there are only 65 cars in the lot. The Strong Pigeonhole Principle guarantees that at least how many people will go in one car together? A. 1 B. 2 C. 3 D. 4 E. 5 Thursday, 4/17/14, Slide #14