5_Inclusion-Exclusion_More_Counting - Discrete Mathematics...

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Discrete Mathematics Dr. Fred Phelps Lecture 5 The Inclusion- Exclusion Principle & More Counting Problems
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LPV 2.3 & Chen 13 The Inclusion-Exclusion Principle This is covered in LPV 2.3 but the lecture is taken from Chen’s book. 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 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 U U 4 5 5 14. S T W + + = + + =
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Chen 13 - The Inclusion-Exclusion Principle S = {1, 2, 3, 4}; T = {1, 3, 5, 6, 7} and W = {1, 4, 6, 8, 9}. Find 1. Note that 2. 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. 3. We compensate by subtracting the number of those elements which belong to more than one of the three sets S , T and W . . S T W U U 4 5 5 14. S T W + + = + + = 14 2 2 2 8. S T W S T S W T W + + - - - = - - - = I I I
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Chen 13 The Inclusion-Exclusion Principle S = {1, 2, 3, 4}; T = {1, 3, 5, 6, 7} and W = {1, 4, 6, 8, 9}. Find Note that 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. We therefore compensate again by adding the number of those elements which belong to all the three sets S , T and W . Then we have the count . S T W U U 8. S T W S T S W T W + + - - - = I I I 8 1 9 S T W S T S W T W S T V + + - - - + = + = I I I UU S T V U U
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The Inclusion-Exclusion Principle A B C A B C A B A C B C A B C = + + - - - + I I I I I U U
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The Inclusion-Exclusion Principle A B C A B C A B A C B C A B C = + + - - - + I I I I I U U
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The Inclusion-Exclusion Principle
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The Inclusion-Exclusion Principle Provin g
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