Ch8.6-Inclusion-exclusion-Applications-part-1.ppt - Applications of the Principle of InclusionExclusion Part I Jorge A Cobb The University of Texas at

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1 Applications of the Principle of Inclusion- Exclusion Part I Jorge A. Cobb The University of Texas at Dallas

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2 Alternative form of Inclusion-Exclusion Assume we have a universe U , | U | = N ( U ) Assume we have n object properties, P 1 , …, P n Example: Universe of cars P 1 = car is sedan P 3 = car is blue P 2 = car is minivan P 4 = car is black Let A i be the subset of U with property P i . A 1 = set of cars that are sedans Let N ( P j 1 P j 2 ... P j k ) the number of elements with properties P j 1 , P j 2 , ..., P j k , i.e., | A j 1 A j 2 ... A j k | E.g. N ( P 1 P 4 ) = number of cars that are black sedans.
Continued …. Let N ( P j 1 P j 2 ... P j k ) the number of elements with properties P j 1 , P j 2 , ..., P j k , i.e., | A j 1 A j 2 ... A j k | Let P i ′ be the negation (opposite) of P i , We are interested in the number of objects without any property 3 ) ( ) 1 ( ) ( ) ( ) ( | | ) ' ' ' ( 2 1 1 1 1 2 1 2 1 n n n k j i k j i n j i j i n i i n n P P P N P P P N P P N P N N A A A N P P P N

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4 The number of onto functions A function f : A→B is onto or surjective iff for every y
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