Inclusion exclusion

Inclusion exclusion - Jim Lambers Math 6A Spring Quarter...

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Jim Lambers Math 6A Spring Quarter 2003-04 Lecture 27 Notes These notes correspond to Section 6.5 in the text. Inclusion-Exclusion In Lecture 7, we learned a useful technique for counting problems called the inclusion-exclusion principle . In its simplest form, the principle states that given two sets A 1 and A 2 , | A 1 A 2 | = | A 1 | + | A 2 | - | A 1 A 2 | . Intuitively, this principle states that the number of ways to complete two tasks, when the tasks can possibly be performed at the same time, is equal to the sum of the number of ways to perform each task, less the number of ways to perform both. An alternative interpretation is that the number of objects that have either one of two properties is equal to the sum of the number of objects that have each property, less the number of objects that have both properties. This principle can easily be generalized to count the number of objects in the union of three sets A 1 , A 2 and A 3 . Using the formula for the cardinality of the union of two sets, along with the Distributive Laws and the Associative Laws, we obtain | A 1 A 2 A 3 | = | A 1 ( A 2 A 3 ) | = | A 1 | + | A 2 A 3 | - | A 1 ( A 2 A 3 ) | = | A 1 | + | A 2 | + | A 3 | - | A 2 A 3 | - | A 1 ( A 2 A 3 ) | = | A 1 | + | A 2 | + | A 3 | - | A 2 A 3 | - | ( A 1 A 2 ) ( A 1 A 3 ) | = | A 1 | + | A 2 | + | A 3 | - | A 2 A 3 | - [ | A 1 A 2 | + | A 1 A 3 | - | ( A 1 A 2 ) ( A 1 A 3 ) | ] = | A 1 | + | A 2 | + | A 3 | - | A 2 A 3 | - [ | A 1 A 2 | + | A 1 A 3 | - | A 1 A 2 A 3 | ] = | A 1 | + | A 2 | + | A 3 | - | A 1 A 2 | - | A 1 A 3 | - | A 2 A 3
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Inclusion exclusion - Jim Lambers Math 6A Spring Quarter...

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