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Unformatted text preview: 19 Inclusion/Exclusion Principle and Pigeonhole Principle 19.1 Inclusion/exclusion principle | A B | = | A | + | B | - | A B | no proof use diagrams to convince them of validity | A B C | = | A | + | B | + | C | - | A B | - | A C | - | B C | + | A B C | again, use diagrams to convince them an example 40 people asked about usage of three drugs: A , B , and C for headaches 23 use A 18 use B 31 use C 11 use A and B 19 use A and C 14 use B and C 37 use at least one how many are using none? how many are using all 3? how many are using exactly one? 4 or more sets show how to generalize 19.2 Basic pigeonhole principle Given n pigeons placed in k < n pigeonholes, at least one pigeonhole contains more than one pigeon note that the principle does not say that any particular pigeonhole will contain anything text proves validity of principle we will take it as an axiom 19.3 Simple examples of basic principle there must be at least two people in first year Engineering with the same birthday at least two students at U of T have the same first and last initials 19.4 Extended principle suppose we want to put 10 objects in 3 containers by principle, we can say that at least one container must have at least two in fact, we can say something stronger at least one container must have at least 4 objects Generally, given n pigeons in k pigeonholes, at least one pigeonhole has at least n k pigeons 19.5 Simple examples of extended principle in a group of 50 people, there must be at least 5 who were born in the same month 40 19.6 More complex examples many problems can be solved using the principle the trick is usually trying to find out what the pigeons and pigeonholes should be an example consider a party of n people, all either mutual acquaintances or mutual strangers i.e. never have a situation in which a knows b but b does not know a (if they want something more practical, processors connected directly to each other)...
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- Fall '06