MATH 245 CH13

# MATH 245 CH13 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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Unformatted text preview: DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1992, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 13 PRINCIPLE OF INCLUSION-EXCLUSION 13.1. Introduction To introduce the ideas, we begin with a simple example. Example 13.3.1. Consider the sets S = { 1 , 2 , 3 , 4 } , T = { 1 , 3 , 5 , 6 , 7 } and W = { 1 , 4 , 6 , 8 , 9 } . Suppose that we would like to count the number of elements of their union S T W . We might do this in the following way: (1) We add up the numbers of elements of S , T and W . Then we have the count | S | + | T | + | W | = 14 . Clearly we have over-counted. For example, the number 3 belongs to S as well as T , so we have counted it twice instead of once. (2) We compensate by subtracting from | S | + | T | + | W | the number of those elements which belong to more than one of the three sets S , T and W . Then we have the count | S | + | T | + | W | - | S T | - | S W | - | T W | = 8 . But now we have under-counted. For example, the number 1 belongs to all the three sets S , T and W , so we have counted it 3- 3 = 0 times instead of once. (3) We therefore compensate again by adding to | S | + | T | + | W | - | S T | - | S W | - | T W | the number of those elements which belong to all the three sets S , T and W . Then we have the count | S | + | T | + | W | - | S T | - | S W | - | T W | + | S T W | = 9 , which is the correct count, since clearly S T W = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Chapter 13 : Principle of Inclusion-Exclusion page 1 of 5 Discrete Mathematics c W W L Chen, 1992, 2008 From the argument above, it appears that for three sets S , T and W , we have | S T W | = ( | S | + | T | + | W | ) | {z } one at a time 3 terms- ( | S T | + | S W |...
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## MATH 245 CH13 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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